How to Calculate Energy Lost: 3 Facts Beginner’s Don’t Know

How to Calculate Energy Lost

Understanding how to calculate energy lost is crucial in various fields, including physics, engineering, and environmental science. Energy loss can occur due to friction, collisions, air resistance, heat transfer, and other factors. In this blog post, we will explore step-by-step methods to calculate energy lost in different scenarios, supported by relevant examples, formulas, and mathematical expressions.

How to Calculate Energy Lost Due to Friction

Understanding Friction and its Impact on Energy

Friction is the resistance force that opposes the motion of an object. When two surfaces come into contact, microscopic irregularities interact, causing friction. As a result, energy is transformed into other forms, such as heat and sound, leading to energy loss.

Step-by-Step Guide to Calculate Energy Lost to Friction

To calculate the energy lost due to friction, we can use the following formula:

$Energy\_Lost = Force \times Distance$

Where:
– $Energy\_Lost$ represents the energy lost due to friction.
– $Force$ is the force opposing the motion.
– $Distance$ is the distance over which the force acts.

Let’s consider an example to illustrate this calculation.

Worked out Example

Suppose a block weighing 10 kg slides a distance of 5 meters on a rough surface with a coefficient of friction of 0.2. To calculate the energy lost due to friction, we can use the formula mentioned earlier:

$Energy\_Lost = Force \times Distance$

First, we need to calculate the force opposing the motion using the formula:

$Force = \text{Coefficient of friction} \times \text{Normal force}$

The normal force is equal to the weight of the block, which can be calculated as:

$Normal\_Force = \text{Mass} \times \text{Gravity}$

Substituting the given values, we get:

$Normal\_Force = 10 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 98 \, \text{N}$

Now, we can calculate the force opposing the motion:

$Force = 0.2 \times 98 \, \text{N} = 19.6 \, \text{N}$

Finally, we can calculate the energy lost due to friction:

$Energy\_Lost = 19.6 \, \text{N} \times 5 \, \text{m} = 98 \, \text{J}$

Therefore, the energy lost due to friction in this example is 98 Joules.

How to Calculate Energy Lost in a Spring

Understanding Potential Energy in a Spring

When a spring is compressed or stretched, it stores potential energy. This energy can be lost due to factors such as damping or external forces acting on the spring.

Steps to Calculate Energy Lost in a Spring

To calculate the energy lost in a spring, we need to consider the change in potential energy. The formula to calculate potential energy in a spring is:

$Potential\_Energy = \frac{1}{2}kx^2$

Where:
– $Potential\_Energy$ is the potential energy stored in the spring.
– $k$ is the spring constant, which represents the stiffness of the spring.
– $x$ is the displacement from the equilibrium position.

To calculate the energy lost, we can find the change in potential energy before and after the event causing energy loss.

Worked out Example

Let’s consider the example of a spring with a spring constant of 100 N/m. If the spring is compressed by 0.2 meters, we can calculate the energy lost.

First, we can calculate the potential energy before compression:

$Potential\_Energy\_Initial = \frac{1}{2} \times 100 \, \text{N/m} \times (0.2 \, \text{m})^2 = 2 \, \text{J}$

Next, let’s assume the spring is compressed further by an additional 0.1 meters. The potential energy after compression is:

$Potential\_Energy\_Final = \frac{1}{2} \times 100 \, \text{N/m} \times (0.3 \, \text{m})^2 = 4.5 \, \text{J}$

To calculate the energy lost, we can find the difference between the initial and final potential energy:

$Energy\_Lost = Potential\_Energy\_Final - Potential\_Energy\_Initial = 4.5 \, \text{J} - 2 \, \text{J} = 2.5 \, \text{J}$

Therefore, the energy lost in this example is 2.5 Joules.

How to Calculate Energy Lost in a Collision

Understanding Collision and Energy Conservation

In a collision, energy can be lost due to deformation, sound production, or heat generation. However, the total energy in a closed system remains constant, as per the principle of energy conservation.

Steps to Calculate Energy Lost in a Collision

To calculate energy lost in a collision, we need to consider the initial and final energy states. The formula to calculate kinetic energy is:

$Kinetic\_Energy = \frac{1}{2}mv^2$

Where:
– $Kinetic\_Energy$ is the energy associated with the object’s motion.
– $m$ is the mass of the object.
– $v$ is the velocity of the object.

By determining the difference between the initial and final kinetic energies, we can calculate the energy lost.

Worked out Example

Let’s consider the example of two cars colliding. Car A with a mass of 1000 kg is initially moving at a velocity of 10 m/s, while Car B with a mass of 1200 kg is initially stationary. After the collision, both cars move at a velocity of 5 m/s.

First, we can calculate the initial kinetic energy of Car A:

$Kinetic\_Energy\_Initial\_A = \frac{1}{2} \times 1000 \, \text{kg} \times (10 \, \text{m/s})^2 = 50,000 \, \text{J}$

Next, we calculate the initial kinetic energy of Car B, which is zero as it is stationary.

The final kinetic energy of both cars can be calculated using the mass and velocity after the collision:

$Kinetic\_Energy\_Final = \frac{1}{2} \times 1000 \, \text{kg} \times (5 \, \text{m/s})^2 + \frac{1}{2} \times 1200 \, \text{kg} \times (5 \, \text{m/s})^2 = 18,750 \, \text{J}$

To calculate the energy lost, we subtract the final kinetic energy from the initial kinetic energy:

$Energy\_Lost = Kinetic\_Energy\_Initial\_A - Kinetic\_Energy\_Final = 50,000 \, \text{J} - 18,750 \, \text{J} = 31,250 \, \text{J}$

Therefore, the energy lost in this collision is 31,250 Joules.

How to Calculate Energy Lost to Air Resistance

Understanding Air Resistance and its Impact on Energy

Air resistance, also known as drag, is a force that opposes the motion of an object through the air. It can cause energy loss by converting the object’s kinetic energy into heat and sound.

Steps to Calculate Energy Lost to Air Resistance

Calculating energy lost to air resistance can be challenging due to the complex nature of the force. However, we can estimate it by comparing the mechanical energy of the system before and after the event causing energy loss.

Worked out Example

Consider a baseball thrown horizontally with an initial velocity of 30 m/s. Due to air resistance, the velocity decreases to 25 m/s after traveling a distance of 50 meters. To calculate the energy lost to air resistance, we can compare the initial and final kinetic energies.

The initial kinetic energy is given by:

$Kinetic\_Energy\_Initial = \frac{1}{2} \times \text{mass} \times (\text{initial velocity})^2$

Substituting the values, we get:

$Kinetic\_Energy\_Initial = \frac{1}{2} \times \text{mass} \times (30 \, \text{m/s})^2$

Next, we calculate the final kinetic energy:

$Kinetic\_Energy\_Final = \frac{1}{2} \times \text{mass} \times (\text{final velocity})^2$

Substituting the values, we get:

$Kinetic\_Energy\_Final = \frac{1}{2} \times \text{mass} \times (25 \, \text{m/s})^2$

To calculate the energy lost, we subtract the final kinetic energy from the initial kinetic energy:

$Energy\_Lost = Kinetic\_Energy\_Initial - Kinetic\_Energy\_Final$

Therefore, the energy lost to air resistance can be determined by calculating the difference in kinetic energy before and after the event, given the initial and final velocities of the object.

How to Calculate Energy Lost to Heat, Sound, and Surroundings

Understanding Energy Conversion to Heat, Sound, and Surroundings

Energy can be lost to the surrounding environment in the form of heat and sound. These losses occur due to various processes like conduction, radiation, and dissipation.

Steps to Calculate Energy Lost to Heat

The amount of energy lost to heat can be calculated using the formula:

$Energy\_Lost\_Heat = \text{Heat Capacity} \times \text{Change in Temperature}$

Where:
– $Energy\_Lost\_Heat$ represents the energy lost to heat.
– $\text{Heat Capacity}$ is the amount of heat required to raise the temperature of a substance by one degree.
– $\text{Change in Temperature}$ is the difference in temperature before and after the event causing energy loss.

Steps to Calculate Energy Lost to Sound

The energy lost to sound can be calculated by subtracting the initial and final sound energies. However, calculating sound energy can be complex and requires specific information about the sound waves produced.

Steps to Calculate Energy Lost to Surroundings

Energy lost to the surroundings can be calculated by considering various factors like radiation, conduction, and dissipation. However, the specific calculations depend on the nature of the event causing energy loss.

Worked out Examples

Let’s consider an example to calculate energy lost to heat. Suppose a 1 kg metal object experiences a temperature increase of 10 degrees Celsius. If the specific heat capacity of the metal is 0.5 J/g°C, we can calculate the energy lost to heat:

First, we need to convert the mass of the object from kg to grams:

$Mass = 1 \, \text{kg} \times 1000 = 1000 \, \text{g}$

Next, we can calculate the energy lost to heat using the formula mentioned earlier:

$Energy\_Lost\_Heat = \text{Heat Capacity} \times \text{Change in Temperature}$

Substituting the given values, we get:

$Energy\_Lost\_Heat = 0.5 \, \text{J/g°C} \times 1000 \, \text{g} \times 10 \, \text{°C} = 5000 \, \text{J}$

Therefore, the energy lost to heat in this example is 5000 Joules.

How to Calculate Energy Lost Between Trophic Levels

Understanding Energy Flow in Ecosystems

In an ecosystem, energy flows from one trophic level to another. However, not all energy is transferred efficiently, and energy losses occur between trophic levels.

Steps to Calculate Energy Lost Between Trophic Levels

To calculate the energy lost between trophic levels, we need to consider the energy transfer efficiency. The formula to calculate energy transfer efficiency is:

$Energy\_Transfer\_Efficiency = \frac{\text{Energy Available at the Next Trophic Level}}{\text{Energy Available at the Previous Trophic Level}} \times 100$

By subtracting the energy transfer efficiency from 100%, we can calculate the energy lost as a percentage.

Worked out Example

Let’s consider an example where the energy available at the previous trophic level is 1000 kcal and the energy available at the next trophic level is 800 kcal. We can calculate the energy transfer efficiency:

$Energy\_Transfer\_Efficiency = \frac{800 \, \text{kcal}}{1000 \, \text{kcal}} \times 100 = 80\%$

To calculate the energy lost, we subtract the energy transfer efficiency from 100%:

$Energy\_Lost = 100\% - 80\% = 20\%$

Therefore, the energy lost between trophic levels in this example is 20%.

By understanding how to calculate energy lost in various scenarios, we can gain valuable insights into the factors that contribute to energy dissipation and optimize energy usage in different systems. Whether it’s analyzing friction, spring energy, collisions, air resistance, or energy flow in ecosystems, the ability to quantify energy losses is an essential skill in many fields. Use the formulas, step-by-step guides, and worked-out examples provided in this blog post as a reference to master the calculation of energy losses.

Numerical Problems on How to calculate energy lost

Problem 1

A car of mass $m = 1000 \, \text{kg}$ is moving with a velocity $v = 20 \, \text{m/s}$ . It collides with a stationary car of mass $M = 1200 \, \text{kg}$ . After the collision, both cars move with a velocity $v' = 10 \, \text{m/s}$ .

Find the initial kinetic energy, final kinetic energy, and the amount of energy lost in the collision.

Solution

Let’s first calculate the initial kinetic energy of the system:

[
KE_{text{initial}} = frac{1}{2} m v^2
]

Substituting the given values:

[
KE_{text{initial}} = frac{1}{2} cdot 1000 , text{kg} cdot $20 \, \text{m/s}$ ^2
]

Simplifying:

[
KE_{text{initial}} = 20000 , text{J}
]

Next, let’s calculate the final kinetic energy of the system:

[
KE_{text{final}} = frac{1}{2} (m + M) v’^2
]

Substituting the given values:

[
KE_{text{final}} = frac{1}{2} $1000 \, \text{kg} + 1200 \, \text{kg}$ cdot $10 \, \text{m/s}$ ^2
]

Simplifying:

[
KE_{text{final}} = 11000 , text{J}
]

Finally, the energy lost in the collision is given by:

[
text{Energy Lost} = KE_{text{initial}} – KE_{text{final}}
]

Substituting the calculated values:

[
text{Energy Lost} = 20000 , text{J} – 11000 , text{J}
]

Simplifying:

[
text{Energy Lost} = 9000 , text{J}
]

Therefore, the amount of energy lost in the collision is $9000 \, \text{J}$ .

Problem 2

A ball of mass $m = 0.5 \, \text{kg}$ is dropped from a height $h = 10 \, \text{m}$ . When it hits the ground, it rebounds with a velocity $v' = -4 \, \text{m/s}$ .

Find the initial potential energy, final kinetic energy, and the amount of energy lost during the bounce.

Solution

The initial potential energy of the ball is given by:

[
PE_{text{initial}} = mgh
]

Substituting the given values:

[
PE_{text{initial}} = 0.5 , text{kg} cdot 9.8 , text{m/s}^2 cdot 10 , text{m}
]

Simplifying:

[
PE_{text{initial}} = 49 , text{J}
]

Since the ball rebounds after hitting the ground, its final kinetic energy is given by:

[
KE_{text{final}} = frac{1}{2} mv’^2
]

Substituting the given values:

[
KE_{text{final}} = frac{1}{2} cdot 0.5 , text{kg} cdot $-4 \, \text{m/s}$ ^2
]

Simplifying:

[
KE_{text{final}} = 4 , text{J}
]

The energy lost during the bounce is given by:

[
text{Energy Lost} = PE_{text{initial}} – KE_{text{final}}
]

Substituting the calculated values:

[
text{Energy Lost} = 49 , text{J} – 4 , text{J}
]

Simplifying:

[
text{Energy Lost} = 45 , text{J}
]

Therefore, the amount of energy lost during the bounce is $45 \, \text{J}$ .

Problem 3

A pendulum of mass $m = 0.2 \, \text{kg}$ is released from a height $h = 5 \, \text{m}$ . At the bottom of its swing, it is caught by a person who exerts a force to stop its motion. The pendulum comes to rest at a height $H = 3 \, \text{m}$ from the ground.

Find the initial potential energy, final potential energy, and the amount of energy lost during the catch.

Solution

The initial potential energy of the pendulum is given by:

[
PE_{text{initial}} = mgh
]

Substituting the given values:

[
PE_{text{initial}} = 0.2 , text{kg} cdot 9.8 , text{m/s}^2 cdot 5 , text{m}
]

Simplifying:

[
PE_{text{initial}} = 9.8 , text{J}
]

Since the pendulum comes to rest at a height $H$ after being caught, its final potential energy is given by:

[
PE_{text{final}} = mgh
]

Substituting the given values:

[
PE_{text{final}} = 0.2 , text{kg} cdot 9.8 , text{m/s}^2 cdot 3 , text{m}
]

Simplifying:

[
PE_{text{final}} = 5.88 , text{J}
]

The energy lost during the catch is given by:

[
text{Energy Lost} = PE_{text{initial}} – PE_{text{final}}
]

Substituting the calculated values:

[
text{Energy Lost} = 9.8 , text{J} – 5.88 , text{J}
]

Simplifying:

[
text{Energy Lost} = 3.92 , text{J}
]

Therefore, the amount of energy lost during the catch is $3.92 \, \text{J}$ .

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