Momentum

Angular Momentum with Respect to a Point: A Comprehensive Guide

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angular momentum with respect to a point

Angular momentum with respect to a point is a fundamental concept in classical mechanics that describes the rotational motion of an object around a specific axis or point. This quantity is conserved in a closed system, meaning that it remains constant unless acted upon by an external torque. Understanding the principles and applications of angular momentum with respect to a point is crucial for students and professionals in various fields, including physics, engineering, and astronomy.

Defining Angular Momentum with Respect to a Point

Angular momentum with respect to a point, denoted as L, is a vector quantity that represents the product of an object’s moment of inertia and its angular velocity about a specific axis or point. The formula for calculating angular momentum in two dimensions is:

L = m × v⊥ × r

Where:
– L is the angular momentum (in kg·m²/s)
– m is the mass of the object (in kg)
– v⊥ is the component of the object’s velocity vector that is perpendicular to the line joining the object to the axis of rotation (in m/s)
– r is the distance from the object to the axis of rotation (in m)

The units of angular momentum are kilogram-square meters per second (kg·m²/s).

The Choice of Axis Theorem

angular momentum with respect to a point

The choice of axis theorem states that the angular momentum of a closed system is conserved, regardless of the chosen axis or point of reference. This means that if the angular momentum of a system is conserved when calculated with one choice of axis, it will be conserved for any other choice of axis. This theorem is particularly useful when analyzing the rotational motion of objects in different frames of reference.

The Theorem of Parallel Axes

The theorem of parallel axes is another important concept in the study of angular momentum with respect to a point. This theorem states that the angular momentum of a rigid body about any axis is equal to the angular momentum of the body about a parallel axis through its center of mass, plus the product of its moment of inertia about the center of mass and the component of the angular velocity vector along the parallel axis.

Mathematically, the theorem of parallel axes can be expressed as:

L = Lcm + I × ω

Where:
– L is the angular momentum about the parallel axis
– Lcm is the angular momentum about the center of mass
– I is the moment of inertia about the center of mass
– ω is the angular velocity about the parallel axis

This theorem is particularly useful when analyzing the rotational motion of rigid bodies, as it allows for the calculation of angular momentum about any point or axis, given the properties of the object and its motion.

Examples and Applications

  1. Rotating Rigid Body: Consider a solid cylinder rotating about its central axis. The angular momentum of the cylinder about its central axis can be calculated using the formula:

L = I × ω

Where I is the moment of inertia of the cylinder about its central axis, and ω is the angular velocity of the cylinder.

  1. Satellite Orbiting a Planet: Imagine a satellite orbiting a planet. The angular momentum of the satellite about the planet’s center can be calculated using the formula:

L = m × v⊥ × r

Where m is the mass of the satellite, v⊥ is the component of the satellite’s velocity vector that is perpendicular to the line joining the satellite to the planet’s center, and r is the distance between the satellite and the planet’s center.

  1. Pendulum Motion: Consider a pendulum swinging about a fixed point. The angular momentum of the pendulum about the fixed point can be calculated using the formula:

L = m × v⊥ × r

Where m is the mass of the pendulum, v⊥ is the component of the pendulum’s velocity vector that is perpendicular to the line joining the pendulum to the fixed point, and r is the distance between the pendulum and the fixed point.

Numerical Problems

  1. A solid sphere of mass 5 kg and radius 0.2 m is rotating about an axis passing through its center with an angular velocity of 10 rad/s. Calculate the angular momentum of the sphere about its center.

Given:
– Mass (m) = 5 kg
– Radius (r) = 0.2 m
– Angular velocity (ω) = 10 rad/s

Moment of inertia of a solid sphere about its center:
I = (2/5) × m × r²
I = (2/5) × 5 kg × (0.2 m)²
I = 0.04 kg·m²

Angular momentum (L) = I × ω
L = 0.04 kg·m² × 10 rad/s
L = 0.4 kg·m²/s

  1. A uniform rod of mass 2 kg and length 1 m is suspended from a fixed point at one end. The rod is struck by a bullet of mass 0.1 kg traveling at 500 m/s. Calculate the angular momentum of the rod about the fixed point.

Given:
– Mass of the rod (m_rod) = 2 kg
– Length of the rod (L) = 1 m
– Mass of the bullet (m_bullet) = 0.1 kg
– Velocity of the bullet (v) = 500 m/s

Moment of inertia of the rod about the fixed point:
I = (1/3) × m_rod × L²
I = (1/3) × 2 kg × (1 m)²
I = 0.667 kg·m²

Velocity of the center of mass of the rod after the collision:
v_cm = (m_bullet × v) / (m_rod + m_bullet)
v_cm = (0.1 kg × 500 m/s) / (2 kg + 0.1 kg)
v_cm = 24.39 m/s

Angular momentum of the rod about the fixed point:
L = I × ω
ω = v_cm / (L/2)
ω = 24.39 m/s / (1 m/2)
ω = 48.78 rad/s

L = I × ω
L = 0.667 kg·m² × 48.78 rad/s
L = 32.55 kg·m²/s

Figures and Data Points

Angular Momentum with Respect to a Point

Figure 1: Illustration of angular momentum with respect to a point.

Table 1: Comparison of angular momentum for different objects and scenarios.

Object/Scenario Angular Momentum (kg·m²/s)
Rotating Sphere 0.4
Orbiting Satellite 32.55
Swinging Pendulum 1.2

Conclusion

Angular momentum with respect to a point is a fundamental concept in classical mechanics that describes the rotational motion of an object around a specific axis or point. Understanding the principles and applications of this quantity is crucial for students and professionals in various fields. This comprehensive guide has provided a detailed overview of the definition, theorems, examples, and numerical problems related to angular momentum with respect to a point, equipping readers with the necessary knowledge and tools to apply this concept in their studies and research.

References

  1. Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics (10th ed.). Wiley.
  2. Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers with Modern Physics (10th ed.). Cengage Learning.
  3. Tipler, P. A., & Mosca, G. (2008). Physics for Scientists and Engineers (6th ed.). W. H. Freeman.
  4. Young, H. D., & Freedman, R. A. (2016). University Physics with Modern Physics (14th ed.). Pearson.
  5. Angular Momentum – an overview | ScienceDirect Topics. (n.d.). Retrieved from https://www.sciencedirect.com/topics/physics-and-astronomy/angular-momentum
  6. 5.1: Angular Momentum In Two Dimensions – Physics LibreTexts. (n.d.). Retrieved from https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Map%3A_University_Physics_I_-Mechanics%2C_Sound%2C_Oscillations%2C_and_Waves(OpenStax)/5%3A_Rotational_Motion_and_Angular_Momentum/5.1%3A_Angular_Momentum_In_Two_Dimensions
  7. Is Angular Momentum truly fundamental? – Physics Stack Exchange. (n.d.). Retrieved from https://physics.stackexchange.com/questions/41291/is-angular-momentum-truly-fundamental
  8. Angular momentum about different points – Stack Overflow. (n.d.). Retrieved from https://stackoverflow.com/questions/11165254/angular-momentum-about-different-points

Torque and Angular Momentum: A Comprehensive Guide for Physics Students

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torque and angular momentum

Torque and angular momentum are fundamental concepts in classical mechanics that describe the rotational motion of objects around an axis. Torque is a measure of the force that can cause an object to rotate, while angular momentum is a measure of the amount of rotation an object has. Understanding these concepts is crucial for analyzing the dynamics of various systems, from simple rigid bodies to complex celestial bodies.

Torque: The Rotational Equivalent of Force

Torque is a vector quantity that represents the rotational force acting on an object. It is defined as the product of the magnitude of the force and the perpendicular distance between the axis of rotation and the line of action of the force. Mathematically, torque can be expressed as:

τ = r × F

where:
– τ is the torque (in Newton-meters, N·m)
– r is the position vector from the axis of rotation to the point of force application (in meters, m)
– F is the force vector (in Newtons, N)
– × represents the cross product operation

The direction of the torque vector is determined by the right-hand rule, which states that if the fingers of the right hand are curled in the direction of rotation, the thumb will point in the direction of the torque vector.

Factors Affecting Torque

The magnitude of the torque depends on the following factors:

  1. Force Magnitude: The greater the force applied, the greater the torque.
  2. Distance from Axis: The greater the distance from the axis of rotation to the line of action of the force, the greater the torque.
  3. Angle between Force and Distance: The torque is maximized when the force is perpendicular to the distance vector (i.e., the angle between them is 90 degrees).

These factors can be used to calculate the torque using the formula:

τ = r × F = r · F · sin(θ)

where θ is the angle between the force vector and the distance vector.

Torque Examples and Applications

  1. Opening a Jar Lid: When you apply a force to the lid of a jar, you create a torque that causes the lid to rotate and open.
  2. Tightening a Nut with a Wrench: The longer the wrench, the greater the torque applied to the nut, making it easier to tighten.
  3. Balancing a Seesaw: The torque created by the weight of a person on one side of the seesaw must be balanced by the torque created by the weight of another person on the other side.
  4. Rotating a Door: The torque created by pushing or pulling on a door handle causes the door to rotate around its hinges.

Angular Momentum: The Rotational Equivalent of Linear Momentum

torque and angular momentum

Angular momentum is a vector quantity that describes the rotational motion of an object. It is defined as the product of the object’s moment of inertia and its angular velocity. Mathematically, angular momentum can be expressed as:

L = Iω

where:
– L is the angular momentum (in kilogram-square meters per second, kg·m²/s)
– I is the moment of inertia (in kilogram-square meters, kg·m²)
– ω is the angular velocity (in radians per second, rad/s)

The moment of inertia is a measure of an object’s resistance to rotational motion and depends on the distribution of the object’s mass around the axis of rotation. For a point mass, the moment of inertia is simply the product of the mass and the square of the distance from the axis of rotation. For more complex objects, the moment of inertia can be calculated using integration or measured experimentally.

Factors Affecting Angular Momentum

The angular momentum of an object depends on the following factors:

  1. Mass Distribution: The more the mass is distributed away from the axis of rotation, the greater the moment of inertia and the angular momentum.
  2. Angular Velocity: The greater the angular velocity, the greater the angular momentum.

These factors can be used to calculate the angular momentum using the formula:

L = Iω

Angular Momentum Examples and Applications

  1. Spinning Top: A spinning top has a high angular momentum, which allows it to remain upright and stable.
  2. Gyroscope: Gyroscopes use the conservation of angular momentum to maintain a stable orientation, which is useful in navigation and stabilization systems.
  3. Satellite Stabilization: Satellites use onboard gyroscopes to maintain their orientation and stability in space, where there are no external forces to keep them stable.
  4. Skating and Gymnastics: Skaters and gymnasts use their angular momentum to perform complex rotations and twists, taking advantage of the conservation of angular momentum.

Advanced Concepts in Torque and Angular Momentum

  1. Conservation of Angular Momentum: In a closed system, the total angular momentum is conserved, meaning that the angular momentum of the system remains constant unless an external torque is applied.
  2. Relationship between Torque and Angular Acceleration: The torque acting on an object is proportional to the object’s angular acceleration, as described by the equation:

τ = Iα

where α is the angular acceleration (in radians per second squared, rad/s²).

  1. Angular Momentum of a System of Particles: The total angular momentum of a system of particles is the vector sum of the angular momenta of the individual particles.
  2. Rotational Kinetic Energy: The rotational kinetic energy of an object is given by the formula:

KE_rot = (1/2) Iω²

where KE_rot is the rotational kinetic energy (in joules, J).

  1. Precession and Nutation: Gyroscopes and other rotating objects can exhibit precession and nutation, which are complex rotational motions that involve the interaction between torque and angular momentum.

Conclusion

Torque and angular momentum are fundamental concepts in classical mechanics that are essential for understanding the rotational motion of objects. By understanding the factors that affect these quantities and the relationships between them, students can develop a deeper understanding of the dynamics of various physical systems, from simple rigid bodies to complex celestial bodies.

References:

  1. Angular Momentum – an overview | ScienceDirect Topics
  2. Classical Angular Momentum: Experiment & Definitions – StudySmarter
  3. 11.2: Torque and Angular Momentum – Physics LibreTexts
  4. Torque and Angular Momentum – HyperPhysics
  5. Rotational Dynamics – Khan Academy

19 Angular Momentum Examples: Detailed Explanations

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This article discusses about angular momentum examples. The term angular simply adds a rotational component in the motion. This article discusses about angular momentum and its examples.

First we shall discuss about linear momentum and its true meaning. Then a simple addition of rotational component in the motion will make it angular momentum. This article will focus more on angular momentum and its examples.

What is linear momentum?

Linear momentum is defined as the product of mass of an object with its velocity. It is essential to find the force exerted by the oject’s motion.

We can say that the rate of change of linear momentum will give us the value of force that will be exerted by that object when it crashes into another object or the force that will be needed to stop it from moving. Linear momentum takes place in a linear motion. Let us study about angular momentum in the next section.

What is angular momentum?

The analog of linear momentum in rotational motion is called as angular momentum. The magnitude of angular momentum can be found using three quantities.

These three quantities are- mass of the object, velocity of the object, radius of the trajectory traced by the object in rotational motion. These three quantities are multiplied by each other as a result of which we get the value of angular momentum. We shall see examples of angular momentum in the next section.

angular momentum examples
Image: Gyroscope

Image credits: anonymous, Gyroskop, CC BY-SA 3.0

Angular momentum examples

The angular momentum is the analog of linear momentum in rotational motion. The radius factor peeps in when we deal with objects following rotational motion.

Let us see some of the examples of angular momentum that are given below-

Ice skater

An ice skater spreads his/her hands to maintain stability while rotating on ice, then he/she come closer in order to increase the rotational speed. This is a clear example of conservation of momentum.

Helicopter propellers

The helicopter propellers are so arranged that effect of one propellers cancels out effect of other propeller. This effect is the net centrifugal force which acts outside due to the action of angular momentum.

Gyroscope

Gyroscopes are used for controlling the orientation of the spacecraft/aircrafts. They spin so fast that the force generated out of this angular momentum makes the gyroscopes stand straight on its axis, any deviation in this axis will mean that the aircraft or spacecraft is making a turn.

Fan blades

Similar to the helicopter blades, fan blades are also arranged in such a manner that they will cancel each other’s effect. We should note that the blades with smaller length rotate faster and the blades with greater length move slower. But the net angular momentum remains conserved.

Rotation of Earth

Earth has an angular momentum of magnitude mvr. Where m is the mass of Earth, V is the velocity of Earth and R the radius of Earth.

A top spinning

When a top spins, it keeps on losing its angular momentum due to friction and air resistance. A spinning top can be referred to as object having angular velocity.

A person sitting on a chair and rotating it

A rotating chair is an example of angular momentum. Here we can also note that momentum will be conserved if the person sitting on the chair tries to spreads his legs or joins his legs. The speed of the chair will be increased or decreased according to the legs position implying that the momentum is conserved.

Rotating a luggage bag

We often get bored while waiting for our flights. Sometimes we start rotating our bags to pass our time. These bags will gain angular momentum once we start rotating it.

Toys

Some toys have rotating elements in them or they themselves rotate. Their rotational motion is possible mainly due to angular momentum that they have gained.

Rolling in flight

When an aircraft performs rolling maneuver, it is supposed to have angular momentum. The direction of the momentum is perpendicular to the motion of the aircraft.

Mixer grinder blades

The blades of mixer grinder rotate to cut the vegetables or fruits placed inside the grinder. These blades move as a result of angular momentum gained by them.

Spinning a cricket ball

When we spin a cricket ball, the ball gains some angular momentum. This way the ball turns when it has impact on the ground, this happens due to force generated due to angular momentum.

Swinging a bat

When we swing a bat to hit the ball, the bat gains angular momentum. The momentum is then transferred to the ball which after impacts goes far away from the batsman.

Ballerina

A ballerina rotates on her toes. The hand positions decide the rotational speed of ballerina. If her hands are spread across then the speed is less and if the hands are closer to her body then she rotates faster.

Stone ties with a thread and rotating it

This is the most common example of rotational motion. Once the thread is subject to rotation, the stone tied with it gains angular momentum.

Spinning a basketball

Some basketball players rotate basketball on their fingers. The basketball rotates as a result of angular momentum gained by it.

Discuss throw

A discuss throw player rotates his/her hand in such a way that the discuss also gains some angular momentum. This angular momentum is then transferred to a linear motion after getting released from hand.

Hammer throw

Similar to discuss throw, hammer also gains angular momentum after the player rotates his hands with hammer in it. The hammer attains linear momentum once it is released from the hands of hammer throw player.

Cycling

The wheels of cycles attain angular momentum which is converted to linear momentum of the cycle itself. This way the momentum is also conserved.

Divers making a 360 degree dive

While diving, sometimes, the divers make 360 degrees loop before entering the water body. This is an example angular momentum gained by the diver while jumping of the diving board.

Also Read:

How to Find Angular Momentum: A Comprehensive Guide

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how to find angular momentum

Angular momentum is a fundamental concept in physics that describes the rotational motion of an object. It is a crucial quantity in understanding the dynamics of rotating systems, from the motion of planets to the behavior of subatomic particles. In this comprehensive guide, we will delve into the intricacies of calculating angular momentum in both … Read more

How to Find Final Momentum After Collision: A Comprehensive Guide

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how to find final momentum after collision

When two objects collide, the final momentum of the system is a crucial quantity to determine. This comprehensive guide will walk you through the step-by-step process of finding the final momentum after a collision using the Impulse-Momentum Theorem and the Conservation of Linear Momentum. We’ll cover various examples, including 2D collisions and inelastic collisions, to … Read more

How to Find Momentum in Circular Motion: A Comprehensive Guide

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how to find momentum in circular motion

In the realm of classical mechanics, understanding the concept of momentum in circular motion is crucial for physics students and enthusiasts alike. This comprehensive guide will delve into the intricacies of calculating angular momentum, a vector quantity that describes the rotational motion of an object. By mastering the formulas and principles presented here, you will … Read more

Is Momentum Conserved in an Elastic Collision: When, Why, How, Detailed Facts and FAQs

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Is momentum conserved in an elastic collision? Certainly yes, the momentum which is the strength of the force and energy that drives the object will be conserved.

Momentum will enhance the mass of an object to move further in motion when the velocity changes. Momentum is the quantity responsible for the object under motion, even the velocity changes.

Generally, momentum is the product of velocity and mass when an object goes into motion. All of this is conserved in an elastic collision only because the mass will not change or by any other means.

In elastic collision, the bodies under collision will have their own kinetic energy and momentum. So when they collide with each other, the momentum will be exchanged and also the kinetic energy.

In this way, when we deep look at objects or bodies under constant motion, when they collide with each other, they will definitely exchange the energies and momentum, so there is conservation happening at the collision before and after.

When we usually talk about the momentum, we generally talk about the strength of the force that is acting upon the object that is under motion.

is momentum conserved in an elastic collision
“CERN / ATLAS Particle Collision” by Ars Electronica Festival is licensed under CC BY-NC-ND 2.0

Why is momentum conserved in an elastic collision?

The one main reason as to is momentum conserved in an elastic collision is that there is basically no loss of anything before and after the collision. Hence it is proven that the elastic collision is one of the proofs that kinetic energy can be conserved at any cost.

Let us take an example and understand this better. There are two sets of ball displaced from a rigid surface and is allowed to swing to and forth. So when the balls of both sides swing together in and out without leaving the constant motion, then it is said to be elastic.

When one ball of one set and the other of the second set swing in opposite directions from their equilibrium positions, respectively, then the kinetic energy is not said to be conserved, and then it becomes an inelastic collision.

In this way, we can make out that the kinetic energy of any system before and after any motion should be conserved only then it is said to be elastic motion. In linear motion, the momentum is conserved until the motion is straight for a long time.

Also, elastic collision will consider only the linear momentum into account and will also act accordingly. In an inelastic collision, the colliding bodies stick to one another and will result in no conservation of momentum in the whole process.

When is momentum conserved in an elastic collision?

Basically, for momentum to be conserved in an isolated system, there must be zero external force acting on that particular system taken into consideration. So there is a loss of momentum and kinetic energy per se.

For a motion to happen, there are several factors that aid in its further movement. One such is the momentum along with the friction and so many. Here all these factors make an impact on the momentum also.

When there is continuous friction, the momentum will definitely be affected by it. So when the path of the motion is rough, the friction will be more, and when the path is smooth, the friction will be smooth and also die out soon.

When these factors like friction and external force affect the momentum, then there will be a change in the momentum of the system. So when the kinetic energy of the colliding bodies are the same before and after the collision, then the momentum of the system is said to be conserved.

Momentum is said to have both magnitude and direction for a body that is under constant motion. Now we need to know how linear momentum is conserved in a system and how it is not.

When a ball is allowed to fall to the ground, it is because of the presence of gravity. So there is no momentum conserved. Because it falls to the ground and comes to contact with the floor so once again, momentum is not conserved o after the rebound, there is a possibility of momentum conservation.

ball 1
“Ball number one” by nudelbach is licensed under CC BY-SA 2.0

Is momentum conserved in a superelastic collision?

Yes, the momentum can be conserved on a superelastic collision. For example, there is a collision, and the potential energy will be indeed converted as kinetic energy, so the kinetic energy will be greater after the collision.

In this process of conversion of potential energy to kinetic energy, there will be a conservation of momentum in it. The momentum will be conserved in all types of elastic collision, whether it is super elastic, perfect collision, or partial collision.

We know that super collision means the kinetic energy will be greater after the collision process. Why does this happen? During a collision of two bodies in an isolated system, there will be an increase in kinetic energy by the conservation of it.

During a particle collision, if it is considered to be a super elastic collision, the potential energy will instantly to kinetic energy. When the particles are rest, they will have potential energy, and when they are put in motion in order to collide, they will have kinetic energy.

So by this way, the kinetic energy will be more after the collision in a super elastic collision. When considering this, we also must know that momentum is simply the amount of force to be applied in order to move the particle in motion when velocity changes.

So momentum will remain the same and is conserved irrespective of the types of elastic collisions considered.

Is momentum conserved in a perfectly elastic collision?

The perfectly elastic collision is the collision in which the body in motion loses neither energy nor momentum in that particular given collision process. When two bodies collide with each other, there is no loss of momentum or energy. This is regarded to be the perfect elastic collision.

Momentum is not lost in this process because the kinetic energy created by the particle in the process remains the same and is not lost. Momentum is the quantity of the force that will have to be applied by the body when the speed is changed.

For example, we consider two particles to collide with each other they are said to have kinetic energy that never will change and will be conserved at the end of the process. Momentum will be conserved because there will be no external force acting on the two bodies that are colliding.

Few instances where we see that in an elastic collision, there will be a conservation of momentum and energy. In billiards, when a ball hits another ball, there is an elastic collision occurring.

It is seen that the kinetic energy and the momentum are transferred to another ball but are not lost. Similarly, when a ball is thrown to the ground and bounces back, there is a net force that is there. So the energy and momentum are instantly conserved.

bill
“billiards” by fictures is licensed under CC BY 2.0

Is momentum conserved in partially elastic collisions?

The momentum meaning the strength of force is not really been conserved in this type of collision. These are the real-time collisions in the world. The colliding bodies rarely stick to each other, and the kinetic energy has been lost.

The partial elastic collision is also regarded as the inelastic collision but with very few deviations. When a ball loses its velocity after the motion comes to rest, there is no energy conserved, and in turn, momentum is also not conserved.

Partially elastic collisions are the most common collision found in the real world. There is a loss of kinetic energy which has been converted from the potential energy will be lost in the form of friction. Also, heat and sound are also lost in the form of energy. So there is no chance for the momentum to be lost.

For example, we consider a bullet fired from the gun. The velocity of the bullet is lost once it hits the target. Since the velocity of the target is not as same as the velocity of the bullet, the process is considered to be the inelastic collision, and the momentum is certainly not conserved.

Similarly, with bow and arrow, the velocity of the arrow does not remain the same after releasing itself from the bow. So when there, the velocity before and after the process does not match, and there will be no conservation of momentum.

Is momentum always conserved in elastic collisions?

Yes, momentum will always be conserved inelastic collision. There is always an exception in some instances like there will be no conservation of momentum and energy in a partial elastic collision.

In all types of collision momentum will be conserved at all costs but energy will be lost but not in elastic collision.

When particles collide with each other, they will undoubtedly exchange kinetic energy with each other. In this way, there is no loss of kinetic energy in the whole collision process.

Like we have seen using so many examples, the momentum will be conserved no matter whichever type of collision it is.

Irrespective of the types of collisions, be it a super elastic or perfect collision, the momentum will be conserved even if there is a loss of energy in the system.

Frequently Asked Questions

What is partially elastic and inelastic collision?

A partially elastic collision is the one where energy is not lost, but a partially inelastic collision is the one where energy is lost.

For instance, two balls collide on each other, and there is a bounce back to the same place. In this process, since the ball bounces back, there is no loss of energy, so it is called a partially elastic collision. There will be a sound heard when the balls collide on and bounce. This sound is nothing but the loss of energy, and therefore, this process is called the partially inelastic collision.

Which determines whether the collision is elastic or inelastic?

When there is no loss of kinetic energy, it is regarded as the elastic collision; otherwise, it is known as an inelastic collision.

When any two-bodied collision occurs, the exchange of the individual kinetic energy occurs there will be no kinetic energy lost. But in an inelastic collision the body or any other object or particle in motion of collision will certainly lose the kinetic energy in the form of heat. Sometime energy will ooze out as sound and friction also.

Also Read:

How to Calculate Momentum of a System: Various Problems and Facts

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momentum of a system 1

In the world of physics, understanding the concept of momentum is crucial. Momentum is a fundamental quantity that helps us describe and analyze the motion of objects and systems. By calculating momentum, we can gain insights into the behavior of physical systems, such as the motion of particles, collisions, and even rotating objects. In this blog post, we will explore how to calculate momentum of a system, delve into special cases, and even touch upon the concept of angular momentum. So, let’s get started!

Calculating Momentum of a System

The Formula for Calculating Momentum

The formula for calculating momentum is simple yet powerful. Momentum is defined as the product of an object’s mass and its velocity. Mathematically, it can be expressed as:

\text{Momentum (p)} = \text{Mass (m)} \times \text{Velocity (v)}

where both mass and velocity are vector quantities. In terms of units, momentum is measured in kilogram-meters per second (kg·m/s).

How to Use the Momentum Formula

how to calculate momentum of a system
Image by Tdadamemd – Wikimedia Commons, Wikimedia Commons, Licensed under CC BY-SA 3.0.
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To calculate the momentum of an object, you need to know its mass and velocity. Let’s consider a simple example. Suppose we have a car with a mass of 1500 kg, moving at a velocity of 30 m/s. To determine its momentum, we can use the formula:

\text{Momentum (p)} = \text{Mass (m)} \times \text{Velocity (v)}

Plugging in the values, we get:

\text{Momentum} = 1500 \, \text{kg} \times 30 \, \text{m/s}

Calculating this, we find that the momentum of the car is 45,000 kg·m/s.

Worked out Examples on Calculating Momentum

Let’s work through a few more examples to solidify our understanding.

Example 1:
Suppose a truck with a mass of 2000 kg is moving at a velocity of 20 m/s. What is its momentum?

Using the formula, we can calculate the momentum as:

\text{Momentum} = 2000 \, \text{kg} \times 20 \, \text{m/s}

The momentum of the truck is therefore 40,000 kg·m/s.

Example 2:
Consider a collision between a car and a stationary object. The car has a mass of 1000 kg and is initially traveling at 15 m/s. During the collision, the car comes to a stop. What is the change in momentum of the car?

To find the change in momentum, we need to calculate the final momentum and subtract the initial momentum. The final momentum can be calculated using the formula:

\text{Momentum} = \text{Mass} \times \text{Velocity}

For the car at rest, its final momentum is given by:

\text{Final Momentum} = 1000 \, \text{kg} \times 0 \, \text{m/s} = 0 \, \text{kg·m/s}

The initial momentum of the car is:

\text{Initial Momentum} = 1000 \, \text{kg} \times 15 \, \text{m/s} = 15,000 \, \text{kg·m/s}

Therefore, the change in momentum is:

\text{Change in Momentum} = \text{Final Momentum} - \text{Initial Momentum} = 0 - 15,000 = -15,000 \, \text{kg·m/s}

The negative sign indicates that the momentum of the car has decreased.

Special Cases in Calculating Momentum

How to Determine the Initial Momentum of a System

When dealing with systems of objects, it is essential to consider the initial momentum of the system before any external forces act upon it. The initial momentum of a system can be calculated by simply adding up the individual momenta of each object in the system.

For example, imagine a system consisting of two objects: a car with a mass of 1000 kg moving at 20 m/s and a truck with a mass of 2000 kg moving at 15 m/s. To determine the initial momentum of the system, we add the momenta of the two objects:

\text{Initial Momentum} = \text{Momentum of Car} + \text{Momentum of Truck} = (1000 \, \text{kg} \times 20 \, \text{m/s}) + (2000 \, \text{kg} \times 15 \, \text{m/s})

Calculating this, we find that the initial momentum of the system is 45,000 kg·m/s.

How to Measure the Final Momentum of a System

The final momentum of a system can be determined in a similar manner to the initial momentum. We add up the individual momenta of each object in the system after any external forces have acted upon it.

Continuing with our previous example, let’s say the car collides with the truck, and both objects come to a stop. The final momentum of the system would then be:

\text{Final Momentum} = \text{Momentum of Car} + \text{Momentum of Truck} = (1000 \, \text{kg} \times 0 \, \text{m/s}) + (2000 \, \text{kg} \times 0 \, \text{m/s})

The final momentum of the system is zero, indicating that the objects have come to a complete stop.

Calculating the Total Momentum of a System Before and After Collision

In scenarios involving objects colliding or interacting with each other, we can examine the conservation of momentum. According to the law of conservation of momentum, the total momentum of a system before a collision is equal to the total momentum after the collision, as long as no external forces act on the system.

To calculate the total momentum before and after a collision, we sum up the individual momenta of all objects in the system.

For instance, consider a collision between a car and a truck. The car has a mass of 1000 kg and is initially traveling at 20 m/s, while the truck has a mass of 2000 kg and is initially moving at 15 m/s. The total momentum before the collision is:

\text{Total Initial Momentum} = \text{Momentum of Car} + \text{Momentum of Truck} = (1000 \, \text{kg} \times 20 \, \text{m/s}) + (2000 \, \text{kg} \times 15 \, \text{m/s})

After the collision, let’s assume the car and truck come to a stop. The total momentum after the collision is:

\text{Total Final Momentum} = \text{Momentum of Car} + \text{Momentum of Truck} = (1000 \, \text{kg} \times 0 \, \text{m/s}) + (2000 \, \text{kg} \times 0 \, \text{m/s})

Remarkably, the total initial momentum is equal to the total final momentum, confirming the conservation of momentum.

Angular Momentum of a System

Understanding Angular Momentum

how to calculate momentum of a system
Image by Olivier Cleynen – Wikimedia Commons, Wikimedia Commons, Licensed under CC0.

In addition to linear momentum, we can also consider angular momentum when dealing with rotating objects or systems. Angular momentum is a measure of an object’s rotational motion and depends on its moment of inertia and angular velocity.

How to Calculate Angular Momentum of a Single Disk System

For a single disk rotating around a fixed axis, the angular momentum can be calculated using the following formula:

\text{Angular Momentum (L)} = \text{Moment of Inertia (I)} \times \text{Angular Velocity (ω)}

The moment of inertia depends on the mass distribution of the object and is specific to each shape. The angular velocity is the rate at which the object rotates.

How to Calculate Angular Momentum of a Two Disk System

When dealing with a system of objects, such as two disks rotating around a fixed axis, we can calculate the total angular momentum by summing up the individual angular momenta of each object.

Let’s say we have two disks with different masses, moments of inertia, and angular velocities. The total angular momentum of the system would be:

\text{Total Angular Momentum} = \text{Angular Momentum of Disk 1} + \text{Angular Momentum of Disk 2}

Calculating the angular momentum for each disk and adding them up will give us the total angular momentum of the system.

Worked Out Examples on Calculating Angular Momentum

momentum of a system 2

To reinforce our understanding of angular momentum, let’s work through a couple of examples.

Example 1:
Consider a disk with a moment of inertia of 0.5 kg·m² and an angular velocity of 4 rad/s. What is its angular momentum?

Using the formula for angular momentum, we can calculate:

\text{Angular Momentum} = \text{Moment of Inertia} \times \text{Angular Velocity}

Plugging in the values, we get:

\text{Angular Momentum} = 0.5 \, \text{kg·m²} \times 4 \, \text{rad/s}

Calculating this, we find that the angular momentum of the disk is 2 kg·m²/s.

Example 2:
Suppose we have two disks in a system. Disk 1 has a moment of inertia of 0.3 kg·m² and an angular velocity of 5 rad/s, while Disk 2 has a moment of inertia of 0.2 kg·m² and an angular velocity of 3 rad/s. What is the total angular momentum of the system?

To find the total angular momentum, we add the individual angular momenta of each disk:

\text{Total Angular Momentum} = \text{Angular Momentum of Disk 1} + \text{Angular Momentum of Disk 2}

Calculating this, we find:

\text{Total Angular Momentum} = (0.3 \, \text{kg·m²} \times 5 \, \text{rad/s}) + (0.2 \, \text{kg·m²} \times 3 \, \text{rad/s})

Simplifying this, we determine that the total angular momentum of the system is 3 kg·m²/s.

How can you calculate the momentum of a system before a collision? Calculating momentum before a collision.

To calculate the momentum of a system before a collision, you need to determine the momentum of each individual object involved in the collision and then add them together. The momentum of an object is given by the product of its mass and velocity. By calculating the momentum of each object before the collision and adding them together, you can determine the total momentum of the system before the collision occurs. This process is explained in more detail in the article Calculating momentum before a collision.

Numerical Problems on how to calculate momentum of a system

Problem 1:

A system consists of two objects with masses of 5 kg and 8 kg, respectively. The velocity of the first object is 4 m/s to the right, while the velocity of the second object is 6 m/s to the left. Calculate the total momentum of the system.

Solution:

The momentum of an object is given by the equation:

p = m \cdot v

where p is the momentum, m is the mass, and v is the velocity.

For the first object:
p_1 = m_1 \cdot v_1 = 5 \, \text{kg} \cdot 4 \, \text{m/s} = 20 \, \text{kg m/s}

For the second object:
p_2 = m_2 \cdot v_2 = 8 \, \text{kg} \cdot (-6 \, \text{m/s}) = -48 \, \text{kg m/s}

The total momentum of the system is the sum of the individual momenta:
p_{\text{total}} = p_1 + p_2 = 20 \, \text{kg m/s} + (-48 \, \text{kg m/s}) = -28 \, \text{kg m/s}

Therefore, the total momentum of the system is -28 kg m/s to the left.

Problem 2:

A system consists of three objects with masses of 2 kg, 3 kg, and 4 kg, respectively. The velocities of the objects are 5 m/s to the right, 2 m/s to the right, and 4 m/s to the left. Calculate the total momentum of the system.

Solution:

For the first object:
p_1 = m_1 \cdot v_1 = 2 \, \text{kg} \cdot 5 \, \text{m/s} = 10 \, \text{kg m/s}

For the second object:
p_2 = m_2 \cdot v_2 = 3 \, \text{kg} \cdot 2 \, \text{m/s} = 6 \, \text{kg m/s}

For the third object:
p_3 = m_3 \cdot v_3 = 4 \, \text{kg} \cdot (-4 \, \text{m/s}) = -16 \, \text{kg m/s}

The total momentum of the system is the sum of the individual momenta:
p_{\text{total}} = p_1 + p_2 + p_3 = 10 \, \text{kg m/s} + 6 \, \text{kg m/s} + (-16 \, \text{kg m/s}) = 0 \, \text{kg m/s}

Therefore, the total momentum of the system is 0 kg m/s.

Problem 3:

A system consists of two objects with masses of 6 kg and 9 kg, respectively. The velocity of the first object is 3 m/s to the left, while the velocity of the second object is 7 m/s to the right. Calculate the total momentum of the system.

Solution:

For the first object:
p_1 = m_1 \cdot v_1 = 6 \, \text{kg} \cdot (-3 \, \text{m/s}) = -18 \, \text{kg m/s}

For the second object:
p_2 = m_2 \cdot v_2 = 9 \, \text{kg} \cdot 7 \, \text{m/s} = 63 \, \text{kg m/s}

The total momentum of the system is the sum of the individual momenta:
p_{\text{total}} = p_1 + p_2 = -18 \, \text{kg m/s} + 63 \, \text{kg m/s} = 45 \, \text{kg m/s}

Therefore, the total momentum of the system is 45 kg m/s to the right.

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Is Momentum Conserved In An Isolated System:Why,When And Detailed Facts And FAQs

by
image 5

When a system of particles is isolated from the rest of the world, the total momentum of the system is conserved. In this article, we will see how is momentum conserved in an isolated system.

The conservation of momentum law is derived from the second and third laws of motion, respectively. The overall momentum of the system has always been conserved in an isolated system. The forces in nature are internal to the system and add up to zero for the entire system. Because the external force on the system is zero, the system’s momentum is conserved.

We will discuss the following points in this article:

is momentum conserved in an isolated system
Momentum

Image Credits: No-w-ay in collaboration with H. Caps, Billard, CC BY-SA 4.0

Momentum may be defined as the quantity of motion experienced by a body under consideration. The momentum of a body is defined mathematically as a function of its mass and the velocity at which it is traveling, respectively. Generally, the isolated system isn’t very precise, but it usually covers the bodies that are being looked at.

Read more on Is Momentum Conserved: When, Why, How, Detailed Facts And FAQs

Now, let us see in detail about how is momentum conserved in an isolated system.

In an isolated system when is momentum conserved?

All of the time, it is conserved in any scenario.

Conserved does not always imply constant. It expresses the fact that something cannot be produced or destroyed at any moment. The condition in which a system is exposed to an external force net force, the momentum of the system changes, but it is conserved. Hence, everything about it is conserved at all times, in all situations.

Why is momentum conserved in an isolated system?

In this case, when there is no force from outside of the body system, the system is referred to as an isolated system.

According to Newton’s second law of motion, because no external force occurs on the system, The change in momentum is equal to zero; hence, the linear momentum is conserved.

dP/dt = 0

image 3

Henceforth, |P|= constant, (as derivative of a constant function is 0)

(In which P=mv=linear momentum is used)

The entire linear momentum of an isolated system, that is, a system that is not subjected to any external forces, is conserved.

image 2
Illustration on is momentum conserved in an isolated system

Frequently Asked Questions (FAQ’s)

Q. What do you mean by linear momentum?

Ans: For the sake of simplicity, Linear Momentum is used to grasp the quantitative idea of motion.

 The linear momentum is a measure of motion that quantifies both velocity and mass. It is defined mathematically as the product of mass and velocity.

It is represented as follow

P = mv

Where, P = Linear momentum

m= mass and v = velocity

Q. What is angular momentum?

Ans: In simple words, it is the momentum of rotating objects.

The term “momentum” refers to the product of the object’s mass and its velocity. Momentum may be found in any object that is moving with mass. The main difference between angular momentum is that it deals with revolving or spinning things.

It is represented as follow

L = mvr

Where, L = Angular momentum

m= mass, v= velocity and r = radius

Q. What do you imply when you speak of conservation of momentum?

Ans: Conservation means no change in the system or one can say that the initial or final value remains the same. The conservation of momentum law could be stated as follow:

“For two or more bodies in an isolated system acting upon each other, their total momentum remains constant unless an external force is applied. Therefore, momentum can neither be created nor destroyed.”

Read more on 17 Momentum Example: Detailed Insight

Q. What is the formula of law of conservation of momentum and what are examples of it?

Ans: The law of conservation of momentum can be represented as follow:

image 5

Where,

The masses of the bodies are denoted by m1 and m2, and the initial velocities of the bodies are denoted by u1 and u2. The final velocities of the bodies are represented by v1 and v2.

Examples of conservation of momentum are given as follow:

  • The collision of two balls or cars
  • The rocket thrust
  • Newton’s cradle

With the help of Newton’s cradle, you can see how the balls in a row of balls will push forward when one ball is raised and then let go of the ball at the other end of the row.

is momentum conserved in an isolated system
Newtons Cradle

Image Credits: The Dean of Physics, Newtons Cradle, CC BY-SA 4.0

Q. In the background, there are automobiles with weight of 5 kilogrammes and 6 kilogrammes, respectively. An automobile with a mass of 5 kg is moving at a velocity of 4 m.s-1 in the direction of the east. Calculate the velocity of an automobile with a mass of 6 kg in relation to the ground.

Ans: Given,

m1 = 5 kg, m2 = 6 kg, v2 = 4 m.s-1, v1 = ?

According to the rule of conservation of momentum, we can say that

Pinitial = 0, because the automobiles are at rest

Pfinal = p1 + p2

Pfinal = m1.v1 + m2.v2

= (5 kg). v1 + (6 kg).  (4 m.s-1)

Pi = Pf

0=(5 kg). v1 + 24 kg.m.s-1

v1 = 4.8 m.s-1

Q. Sedans of 100 kg and 200 kg mass are at rest. A 200 kilogramme car travelling at 60 m.s-1towards the west. Find the car’s velocity relative to the land.

Ans: Given,

m1 = 100 kg, m2 = 200 kg, v2 = 60 m.s-1, v1 = ?

When we look at the law of momentum, we can see that it holds true that,

Pinitial = 0, as the sedans are at rest

Pfinal = p1 + p2

Pfinal = m1.v1 + m2.v2

= (100 kg). v1 + (200 kg).  (60 m.s-1)

Pi = Pf

0=(100 kg). v1 + 12000 kg.m.s-1

v1 = 120 m.s-1

Therefore, the sedan car velocity relative to the ground is 120 m.s-1

Q. What is the meaning of an isolated system?

Ans: There must be two or more objects in order for a system to exist. For example, a system that is completely isolated from external influences is referred to as an isolated system.

A system that is isolated is one in which the only forces that contribute to the change in momentum of an individual item are the forces operating between the components themselves, and hence the object is not affected by any external forces.

There are two things that must happen for there to be a net external force:

  • There is an external force present in the system that originates from a source other than the two objects of the system.
  • A force that is unbalanced in relation to other forces.

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