## Table of Contents

**Centripetal force**: the thing that keeps you from flying off the edge of a merry-go-round like your hopes and dreams! It is a fundamental concept in the study of circular motion. This force is directed toward the center of the circle to keep an object moving on a curved path. It can be provided by things like tension in a string or gravitational attraction.

On the other side, **centrifugal force** isn’t really a force at all! It’s an apparent force that seems to push objects away from the center. In reality, there’s no outward force really acting on an object in a circular motion. The sensation we feel is just our inertia trying to keep us going in a straight line.

Plus, **Newton’s third law of motion** states that for every action, there is an equal and opposite reaction. In this case, the centripetal force pulling inward is balanced by the centrifugal force pushing outward. Interesting, right?

**Centripetal Force**

Behold the following table, displaying some major aspects of Centripetal Force:

Concept | Centripetal Force |

Definition | A force that pulls an object to the center of its curved path |

Direction | Always towards the center |

Examples | Ball on a string swung in circles |

It’s important to remember that Centripetal Force is necessary for maintaining circular movement. If it wasn’t there, objects would keep going in a straight line, according to Newton’s 1st law of motion.

**Fun Fact:** Centripetal Force was initially described in Sir Isaac Newton’s book “*Mathematical Principles of Natural Philosophy*“. Plus, centrifugal force is the perfect excuse for things going off the rails.

**Centrifugal Force**

**Centrifugal force is an apparent force that pushes objects away from the center of rotation when moving in a curved path.** It is actually due to inertia and the tendency of the object to continue on a straight line. This sensation counters the **centripetal force, which pulls the object toward the center of rotation**.

The table below shows some key aspects of centrifugal force:

Property | Description |

Definition | The apparent force that pushes away from the center of rotation |

Direction | Away from the center of rotation |

Cause | Inertia and tendency to continue in a straight line |

Interaction | Counteracts the centripetal force |

Relationship | Opposite to centripetal force |

Impact | Creates a feeling of being pushed outwards from the center of rotation |

Circular motion involves more than just centrifugal and centripetal forces. Other factors such as centripetal acceleration and gravitational forces also play a part.

To better understand this concept, it is important to clarify misconceptions, emphasize centripetal forces, provide visual aids, and relate to everyday examples. **Centripetal and centrifugal forces can be thought of as dysfunctional couple, always pulling in different directions and causing confusion in a circular motion.**

**Difference Between Centripetal and Centrifugal Forces**

Centripetal and centrifugal forces are two terms used in physics to explain circular motion. Even though they may seem similar, they have different roles and effects.

Force Type | Description | Examples |

Centripetal Force | Always acts towards the center of a circular path, necessary to keep an object moving in a circle. | Tension in a string, gravitational force, friction |

Centrifugal Force | Pseudo-force create the illusion of being pushed away from the center of rotation, not a real force. | the feeling of being pushed outwards while turning |

The historical facts related to these forces are also interesting. Sir Isaac Newton first presented these concepts in his ‘Mathematical Principles of Natural Philosophy’ published in 1687. His laws of motion and gravity revolutionized our understanding of motion.

So, remember: **centripetal force is the one that keeps us on track, while centrifugal force is the rebel.**

**Circular Motion and Forces**

Circular motion and forces are intricately connected. Forces are crucial for maintaining the motion of objects. Two opposing forces arise: **centripetal and centrifugal**. **The centripetal force acts towards the center and keeps the object on its path. The centrifugal force appears outward from the center, seeming to push the object away.** However, it is important to note that this is an apparent force due to inertia.

Let’s look at the table:

Forces | Centripetal Force | Centrifugal Force |

Meaning | Force-directed toward the center of the circular path | Apparent force pushing outward from the center of circular motion |

Direction | Always inward | Always outward |

Examples | Tension in a pendulum | The feeling of being pushed outwards on a rotating carousel |

Gravitational force keeping planets in orbit around the sun |

**Centrifugal force is not a real force. It is merely an apparent force from inertial effects.**

Centripetal force can be seen in many natural phenomena and daily experiences. For example, friction between car tires and the road provides centripetal force when driving around curves. The tension in a rope can provide a centripetal force to keep a bucket of water moving in a vertical circle without spilling.

A true story further highlights the importance of forces in a circular motion. A group of friends went to an amusement park and rode a roller coaster with twists and turns. They felt the powerful forces acting on their bodies. The feeling of being pushed outward during sharp turns was a result of centrifugal force. And the discomfort when accelerating or decelerating was due to centripetal force. This experience gave them a newfound appreciation for the interplay between circular motion and forces.

Be ready to unravel the mysterious dance between real forces and the tricks of apparent forces.

**Real and Apparent Forces**

Real and apparent forces are concepts that help us understand the motion of objects in circles. Let’s look at the difference between them:

**Real Force:** Exists, can be measured directly, and acts on objects due to contact. Examples include gravitational, frictional, and electromagnetic forces.

**Apparent Force:** Seems to exist, cannot be measured directly, and arises from our frame of reference. Examples include centrifugal and Coriolis forces.

**Apparent forces only appear when we observe an object from a rotating or accelerating frame. They are not physically present, just perceived due to our motion.**

In short, real forces are tangible and measurable, whereas apparent forces come from our perspective. Knowing this helps us analyze circular motion better.

**Pro Tip:** Remember to factor in both real and apparent forces when analyzing circular motion or rotating frames of reference. Centripetal and centrifugal forces, the dynamic duo of circular motion, keep us spinning and reeling, just like a wild salsa party.

**Examples of Centripetal and Centrifugal Forces**

Isaac Newton introduced the concept of **centripetal force** in his book “Philosophiæ Naturalis Principia Mathematica.” His laws of motion laid the basis for understanding both centripetal and centrifugal forces. These forces play an important role in circular motion. To understand them better, let’s look at some examples:

Example | Centripetal Force | Centrifugal Force |

Planets revolving around the sun | A gravitational force keeps them in orbits. | An outward force due to inertia. |

A car turning a corner | Friction between tires and the road. | Pushed against door due to inertia. |

Spinning a yo-yo around your finger | Tension in string creates the centripetal force. |

The battle between centripetal and centrifugal forces is like balancing or going for a spin – either way, it’ll leave you feeling a bit dizzy.

**Centripetal Acceleration and Centrifugal Acceleration**

Centripetal acceleration can be explained as, it is a property of a body that is moved by a path in a circular way while centrifugal acceleration can be explained as, a property of a matter which is moved by a circular way and it is straight externally the path of the circle.

**Formula:-**

The formula for the centrifugal acceleration is the same as the centripetal acceleration. The formula for the centripetal acceleration and centrifugal acceleration is the product of tangential velocity squared and mass, which is divided by the radius that signifies that on doubling that tangential velocity, the centripetal acceleration and centrifugal acceleration will be quadrupled.

**Image – A body experiencing ****uniform circular motion**** requires a centripetal force, towards the axis as shown, to maintain its circular p**ath;

**Image Credit – ****Wikipedia**

Mathematically the expression for the centripetal acceleration and centrifugal acceleration can be written as

F = mv2/r

Where

F represents as Centrifugal acceleration and the unit used to measure is meters per second square.

m represents as Mass of the matter and the unit used to measure is kilogram.

v represents, Speed or velocity of the product, and the unit used to measure is meters per second square.

r represents radius and the unit used to measure is radians.

**Similarities between centripetal acceleration and centrifugal acceleration:-**

The common things between centripetal acceleration and centrifugal acceleration are listed below,

- The both properties of centripetal acceleration and centrifugal acceleration depend on the moving objects which are rotated in a circular path.
- When force is applied to an object from externally at that time centripetal acceleration and centrifugal acceleration are generated.

**Is centripetal acceleration and centrifugal acceleration the same?**

No, centripetal acceleration and centrifugal acceleration are not the same terms. Centripetal acceleration always points towards the center of the circle and centripetal acceleration is straight externally the path of the circle.

**When centripetal acceleration and centrifugal acceleration are same?**

Centripetal acceleration and centrifugal acceleration are able to act at the same time because to generate the centripetal acceleration centripetal force is required at the same time to generate the centrifugal acceleration centrifugal force is required, both forces never can be acted at the same time on a moving object.

**Characteristics of Centripetal acceleration:-**

The characteristics of the centripetal or radial acceleration are listed below,

- The characteristics of the motion of the pendulum traversing a path in a circular way, and centripetal acceleration are always notified according to the center of the path in a circular way.
- The magnitude of the centripetal or radial acceleration can be expressed as,

- The direction for the radial or centripetal acceleration is all time changes.
- For, U.C.M. the magnitude of the centripetal or radial acceleration is unchanged.
- The centripetal or radial acceleration is identified as a letter. S.I. unit to measure the centripetal or radial acceleration is a meter per seconds square.
- The centripetal or radial acceleration is always conducted towards the spot of the circular way along the radius.

**What is the relation between centripetal acceleration and centrifugal acceleration?**

Centripetal acceleration is constantly spotted along the spot of the circle, in this case, the direction of the moving body all the time changes, and the velocity or speed constantly tangent to the circle while the centrifugal force is an imaginary force which is the matter is felled when it is rotated in a motion through a circular path.

Mathematically the expression for the centripetal acceleration and centrifugal acceleration can be written as

**ar = mv2/r**

Where

F represents Centrifugal acceleration and the unit used to measure is meters per second square.

m represents as Mass of the matter and the unit used to measure is kilogram.

v represents, Speed or velocity of the product, and the unit used to measure is meters per second square.

r represents radius and the unit used to measure is radians.

**Formula derivation for the centripetal acceleration and radial acceleration:-**

A substance name M is attached with a string and created to spin around about a particular permanent spot O which is denoted as the center of the spot. When the substance start to spin around fast the string that time it almost like the radius of the circle. This signifies that a force is acting on the substance from the spot of the circle. For this reason, an acceleration a0 is together with the direction of the radial. (Together with the radius of the circle near the spot of the circle).

To determine this force, tension is generated towards the string in the direction of the opposite. This force for the tension is derived as, centripetal force.

For this reason, the acceleration developed on the substance is called centripetal acceleration or radial acceleration and denoted as ar. Performing the property for the similar triangles and we can write,

A and B both are almost close so we can derive this AB to the length of arc AB and the expression can be written as,

AB = v *dt

figure (3) we can observe A and B almost same and the expression we can write as,

v + dv≅ dv

v *dt/r= dv/v

On rearranging,

dv/dt = v2/r

Thus,

dv/dt

Under the uniform circular motions the centripetal acceleration or radial acceleration is generated and we can write the formula for the centripetal acceleration or radial acceleration,

ar = v2/r

**Image – The velocity vectors at time ***t*** and time ***t*** + ***dt*** are moved from the orbit on the left to new positions where their tails coincide, on the right. Because the velocity is fixed in magnitude at ***v*** = ***r*** ***ω***, the velocity vectors also sweep out a circular path at angular rate ***ω***. As ***dt*** → 0, the acceleration vector a becomes perpendicular to v, which means it points toward the center of the orbit in the circle on the left. Angle ***ω*** ***dt*** is the very small angle between the two velocities and tends to zero as ***dt*** → 0;**

**Image Credit – ****Wikipedia**

**What is the difference between centripetal acceleration and centrifugal acceleration?**

The major difference between centripetal acceleration and centrifugal acceleration is listed below,

Centripetal acceleration | Centrifugal acceleration |

Centripetal acceleration can be explained as; it is a property of a body that is moved by a path in a circular way. | Centrifugal acceleration can be explained as, a property of a matter which is moved in a circular way and is straight externally the path of the circle. |

The direction of the centripetal acceleration is inward. | The direction of the centrifugal acceleration is outward. |

Centripetal acceleration is generated for the reason of centripetal force. | Centrifugal acceleration is generated by the reason of centrifugal force. |

**Problem:1**

**A ball that is made with iron material has a mass of 2 kilograms. The iron ball is connected with a string. When the external force is applied to the ball, in that case, the ball starts to move in a circular motion which follows a circular way. The radius of the circle is 70 centimeters.**

**Now**

**Determine the centripetal acceleration if force is applied in every round for 4 seconds.**

**Determine the amount of force that is applied to the ball to generate centripetal acceleration.**

**Solution:-**

Given the data are,

Mass of the ball (m) = 2 kilogram

Radius (r) = 70 centimeters = 0.7 meters

So, the velocity for the path in which the ball is rated,

v = 1.09 meters per second

Now, we know the formula for centrifugal acceleration,

m/s2

When the centripetal acceleration is generated in that case the centripetal force is applied.

The formula for the centripetal force is,

kilogram meter per second square

A ball which is made with iron material has a mass of 2 kilograms. The iron ball is connected with a string. When the external force is applied to the ball, in that case, the ball starts to move in a circular motion which follows a circular way. The radius of the circle is 70 centimeters. Now

The centripetal acceleration if force is applied in every round for 4 seconds is 1.697 meters per second.

The amount of force that is applied to the ball to generate centripetal acceleration is 3.394-kilogram meters per seconds square.

**Problem:2**

**A toy whose mass is 5 kilograms is attached with a string and spinning round in a circular way continuously. The height of the string with which the toy is connected is 2.2 meters and when the toy is spinning around 300 revolutions per minute.**

**Determine,**

**A. linear velocity of the toy.**

**B. Acceleration of the toy.**

**C. The force is applied to the toy.**

**Solution**:-

Given data are,

m = 5-kilogram

r = 2.2 meters

N = 300 revolutions per minute

We know that

v = 69.08 meter per second

a = v2/r

a = 69.082/2.2

a = 2169 meter per second square.

Centripetal force,

F = ma

F = 5*2169

F = 10845 Newton

A toy whose mass is 5 kilograms is attached with a string and spinning round in a circular way continuously. The height of the string with which the toy is connected is 2.2 meters and when the toy is spinning around 300 revolutions per minute.

A. linear velocity of the toy is 69.08 meters per second.

B. Acceleration of the toy is 2169 meters per second square.

C. The force applied upon the toy is 10845 Newton

**Frequently Asked Questions**

**Q: What is the difference between centripetal force and centrifugal force?**

A: Centripetal force is the inward force acting on an object moving in a circular path, whereas centrifugal force is the outward force acting on the same object in the opposite direction.

**Q: How does centrifugal force differ from centripetal force?**

A: Centripetal force always acts perpendicular to the direction of motion, while centrifugal force acts parallel to the direction of motion.

**Q: Can you explain the concept of centripetal vs centrifugal force?**

A: Centripetal force is the force that is directed towards the center of a circular path, while centrifugal force is the apparent outward force felt by an object moving in a circular path.

**Q: What is an example of centripetal force?**

A: When a car takes a turn, the force exerted by the tires towards the center of the turn is an example of centripetal force.

**Q: What is an example of centrifugal force?**

A: When a spinning top wobbles and tilts, the apparent outward force pulling the top away from the center is an example of centrifugal force.

**Q: What is the relationship between centripetal force and centrifugal force?**

A: Centripetal force and centrifugal force are opposite and equal forces that act on an object in a circular motion.

**Q: Is centrifugal force a real force?**

A: Centrifugal force is not a real force but an apparent force that is felt by an object moving in a circular path.

**Q: How is centripetal force provided?**

A: Centripetal force is provided by the component of force acting on the object perpendicular to the direction of motion.

**Q: What is the role of the normal force in centripetal vs centrifugal force?**

A: Normal force is the force that acts perpendicular to the surface of contact between two objects and provides the necessary centripetal force for circular motion.

**Q: What is an example of centrifugal force in everyday life?**

A: The force felt when riding a merry-go-round or a roller coaster is an example of centrifugal force.

**Conclusion**

Centripetal and centrifugal forces are essential to understanding circular motion. **The centripetal force is directed toward the center of the circle, keeping an object moving. The centrifugal force, on the other hand, is an apparent force. It counteracts the centripetal, so the object does not fly off in a straight line. **These forces actually go together.

It is important to know that **centrifugal force does not exist**. It is simply our perception of an outward force because of inertia. When in a rotating frame of reference, we feel the centrifugal force. But, objectively, it does not really exist. The centripetal force, however, always exists. It is the force that keeps an object moving in a circle.

To explain this concept, think of a merry-go-round. As you spin, you feel like the centrifugal force is pushing you outwards. Really, though, it is your body wanting to move in a straight line due to inertia. The centripetal force from the platform of the merry-go-round is what keeps you in a circular motion. It is like a ‘pull’ that maintains the circular motion.