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3.6 Centripetal Acceleration and Centripetal Force

7 min readjanuary 20, 2023

Peter Apps

Peter Apps

Kashvi Panjolia

Kashvi Panjolia

Peter Apps

Peter Apps

Kashvi Panjolia

Kashvi Panjolia

Important Equations

The is equal to the rate of change of with time, and is equal to the rate of change of position with time.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%2011.58.03%20AM.png?alt=media&token=3758dc0c-2a05-4773-82ce-e58711a6c253

Equation:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202023-01-20%20at%209.45-betkexpnH0KJ.png?alt=media&token=5b8e82d8-4502-4046-b0db-969f5cb3b78c

Surprisingly, in , in which an object has constant speed, there is an due to a change in direction. Don’t forget, is a change in , which is a vector quantity that has both magnitude and direction!

Centripetal Acceleration

is the that an object experiences when it moves in a circular path. This is directed toward the center of the circle. It is also known as .

Imagine you are on a roller coaster and you experience the feeling of being pulled toward the center of the loop as you go through a loop-the-loop. This is an example of . The force that is pulling you toward the center of the loop is the .

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-nvYTn7xFIldd.png?alt=media&token=d889f380-28e1-441f-96ad-ba9776d54c14

Image courtesy of Quora.

Force and are both quantities that have both magnitude and direction. The direction of the acting on an object is the same as the direction of its . This means that if you want to change the direction of an object's , you need to change the direction of the acting on it. For example, if you want to change the direction of a car's so it is accelerating and turning right, you need to give it a rightward .

The center of mass (COM) of a system is a point that represents the average position of all the mass in the system. Conceptually, it is the point where the entire mass of the system can be considered to be concentrated, which makes it the point at which the system would balance if it were suspended by a single point. The position of the center of mass depends on the distribution of mass within the system.

Consider a seesaw in a playground. The center of mass of a seesaw is located at the point where the board of the seesaw balances. This point is also known as the . If you place a heavier person on one side of the seesaw, the center of mass will shift closer to that side, and the seesaw will tilt in that direction.

https://imgix.albert.io/user-assets/Wes/c77cbb8a-3a1c-46be-b826-1701f447bf2d-Physics-Fulcrum_3.png?w=0.15&ixlib=js-2.1.2

Image courtesy of Albert.io.

The is the of the center of mass of the system, and it is equal to the rate of change of position of the center of mass with time. The of the center of mass is the rate of change of the with time.

For example, if you want to know how fast a car is moving, you can measure its . If you want to know how fast the car's is changing, you can measure its . Similarly, if you want to know how fast the center of mass of a system is moving, you can measure its , and if you want to know how fast the is changing, you can measure the of the center of mass.

Velocity

While the points towards the center of the circle, the vector does not. A key element of is that the is kept constant, so in order to have due to a change in direction only, the vector must be to the vector at all points along the circle. means that the vector makes a 90-degree angle with the vector when the vectors are placed tail to tail.

https://homework.study.com/cimages/multimages/16/20191221_12473560231609063748.png

Image courtesy of Study.com.

Since the and vectors make a 90-degree angle, there is no component of the that is acting in the same direction as the vector, so the will not affect the magnitude of the . However, the vector does affect the direction of the vector. If you add the and vectors using , you will find that the direction of the is always pointing along the circle, causing .

We can relate the to the tangential using this equation:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202023-01-20%20at%209.45-betkexpnH0KJ.png?alt=media&token=5b8e82d8-4502-4046-b0db-969f5cb3b78c

where a is the , v is the , and r is the radius of the circle.

Centripetal Force

The is defined as the force that is required to make an object move in a circular path. It is equal to the mass of the object multiplied by the . The formula for is: F = m * a_c, where F is the , m is the mass of the object and a_c is the .

Recall that the is not a new force; it is just another name for the directed toward the center of the circle. This could be caused by the normal force, tension, , , or another type of force.

For example, when a car takes a turn, the tires exert on the road, and the force is what provides the to keep the car moving in a circular path. Similarly, when a planet is orbiting the sun, it is the of the sun that acts as the .

The direction of the vector is always pointed toward the center of the circle, like the vector. This occurs because the accleration vector always points in the same direction as the vector (in this case, the ) due to Newton's Second Law (F=ma).

It's also important to note that the greater the speed of an object, the greater the required to keep it moving in a circular path. Similarly, the smaller the radius of a circular path, the greater the required to keep an object moving in that path.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%2011.58.44%20AM.png?alt=media&token=10083a60-4ccf-4f13-b6ab-41fa1338a205

Image courtesy of quickmeme.com.

Practice Questions

1. What is the direction of the of an object undergoing ? A) Radial B) Tangential C) Perpendicular D) Horizontal

Answer: A) Radial

2. An object of mass 5 kg is moving in a circular path of radius 3 m with a constant of 4 m/s. What is the of the object? A) 12 m/s^2 B) 16 m/s^2 C) 20 m/s^2 D) 5.33 m/s^2

Answer: D) 5.33 m/s^2 Explanation: To find the , we use the formula a_c = v^2/r, where a_c is the , v is the of the object and r is the radius of the circular path. Substituting the given values, we get a_c = (4 m/s)^2 / 3 m = 16 m/s^2 / 3 m = 5.33 m/s^2

3. A ball of mass 1 kg is tied to a string and is moving in a circular path of radius 0.5 m. If the ball is moving with a of 3 m/s, what is the acting on the ball? A) 2.25 N B) 4.5 N C) 9 N D) 1.5 N

Answer: D) 1.5 N Explanation: The acting on an object in is given by the formula F = m * a_c, where F is the , m is the mass of the object and a_c is the . To find the , we use the formula a_c = v^2/r, where a_c is the , v is the of the object and r is the radius of the circular path. By substituting the given values, we get a_c = (3 m/s)^2 / 0.5 m = 9 m^2/s^2 / 0.5 m = 18 m/s^2Then we can use the formula F = m * a_c to find the acting on the ball, we know the mass of the ball is 1 kg, so we substitute the values into the formula: F = m * a_c = 1 kg * 18 m/s^2 = 18 N

Key Terms to Review (17)

Acceleration

: Acceleration refers to the rate at which an object's velocity changes over time. It can be positive (speeding up), negative (slowing down), or zero (constant speed).

Center of Mass (COM)

: Center of mass refers to the point where all mass can be considered concentrated for purposes of calculating motion. It represents the average position of all particles comprising an object.

Center of Mass Velocity

: Center of mass velocity refers to the velocity of an object's center of mass, which represents its average motion.

Centripetal Acceleration

: Centripetal acceleration is the acceleration experienced by an object moving in a circular path. It is always directed towards the center of the circle and its magnitude can be calculated using the formula a = v^2/r, where v is the velocity of the object and r is the radius of the circle.

Centripetal Force

: Centripetal force is the net force acting towards the center of an object's circular path. It keeps objects moving in uniform circular motion by continuously changing their direction but not their speed.

Friction

: Friction is a force that opposes relative motion between two surfaces in contact. It arises due to microscopic irregularities between surfaces and can cause objects to slow down or come to rest.

Gravitational Force

: Gravitational force is the attractive force between two objects with mass. It depends on the masses of the objects and the distance between them.

Net Force

: The net force is the overall force acting on an object, taking into account both magnitude and direction. It determines the object's acceleration or deceleration.

Newton's Second Law (F=ma)

: Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In simpler terms, it explains how the motion of an object changes when a force is applied to it.

Pivot Point

: The pivot point is the fixed point around which an object rotates or turns.

Radial Acceleration

: Radial acceleration refers specifically to acceleration directed towards or away from the center of rotation. It occurs when there is either speeding up or slowing down while moving along a curved path.

Resultant Vector

: The resultant vector is the sum of two or more vectors. It represents the combined effect of all individual vectors.

Tangent

: In physics, tangent refers to a line that touches but does not cross or intersect with another curve at one specific point.

Uniform Circular Motion

: Uniform circular motion refers to an object moving along a circular path with constant speed. The object experiences acceleration towards the center of the circle, known as centripetal acceleration.

Vector Addition

: Vector addition is the process of combining two or more vectors to find their resultant vector. It involves adding the magnitudes and directions of the vectors.

Vector Quantity

: A vector quantity is a physical quantity that has both magnitude and direction. It can be represented by an arrow, where the length of the arrow represents the magnitude and the direction of the arrow represents the direction.

Velocity

: Velocity refers to the rate at which an object changes its position in a specific direction. It includes both speed and direction.

3.6 Centripetal Acceleration and Centripetal Force

7 min readjanuary 20, 2023

Peter Apps

Peter Apps

Kashvi Panjolia

Kashvi Panjolia

Peter Apps

Peter Apps

Kashvi Panjolia

Kashvi Panjolia

Important Equations

The is equal to the rate of change of with time, and is equal to the rate of change of position with time.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%2011.58.03%20AM.png?alt=media&token=3758dc0c-2a05-4773-82ce-e58711a6c253

Equation:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202023-01-20%20at%209.45-betkexpnH0KJ.png?alt=media&token=5b8e82d8-4502-4046-b0db-969f5cb3b78c

Surprisingly, in , in which an object has constant speed, there is an due to a change in direction. Don’t forget, is a change in , which is a vector quantity that has both magnitude and direction!

Centripetal Acceleration

is the that an object experiences when it moves in a circular path. This is directed toward the center of the circle. It is also known as .

Imagine you are on a roller coaster and you experience the feeling of being pulled toward the center of the loop as you go through a loop-the-loop. This is an example of . The force that is pulling you toward the center of the loop is the .

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-nvYTn7xFIldd.png?alt=media&token=d889f380-28e1-441f-96ad-ba9776d54c14

Image courtesy of Quora.

Force and are both quantities that have both magnitude and direction. The direction of the acting on an object is the same as the direction of its . This means that if you want to change the direction of an object's , you need to change the direction of the acting on it. For example, if you want to change the direction of a car's so it is accelerating and turning right, you need to give it a rightward .

The center of mass (COM) of a system is a point that represents the average position of all the mass in the system. Conceptually, it is the point where the entire mass of the system can be considered to be concentrated, which makes it the point at which the system would balance if it were suspended by a single point. The position of the center of mass depends on the distribution of mass within the system.

Consider a seesaw in a playground. The center of mass of a seesaw is located at the point where the board of the seesaw balances. This point is also known as the . If you place a heavier person on one side of the seesaw, the center of mass will shift closer to that side, and the seesaw will tilt in that direction.

https://imgix.albert.io/user-assets/Wes/c77cbb8a-3a1c-46be-b826-1701f447bf2d-Physics-Fulcrum_3.png?w=0.15&ixlib=js-2.1.2

Image courtesy of Albert.io.

The is the of the center of mass of the system, and it is equal to the rate of change of position of the center of mass with time. The of the center of mass is the rate of change of the with time.

For example, if you want to know how fast a car is moving, you can measure its . If you want to know how fast the car's is changing, you can measure its . Similarly, if you want to know how fast the center of mass of a system is moving, you can measure its , and if you want to know how fast the is changing, you can measure the of the center of mass.

Velocity

While the points towards the center of the circle, the vector does not. A key element of is that the is kept constant, so in order to have due to a change in direction only, the vector must be to the vector at all points along the circle. means that the vector makes a 90-degree angle with the vector when the vectors are placed tail to tail.

https://homework.study.com/cimages/multimages/16/20191221_12473560231609063748.png

Image courtesy of Study.com.

Since the and vectors make a 90-degree angle, there is no component of the that is acting in the same direction as the vector, so the will not affect the magnitude of the . However, the vector does affect the direction of the vector. If you add the and vectors using , you will find that the direction of the is always pointing along the circle, causing .

We can relate the to the tangential using this equation:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202023-01-20%20at%209.45-betkexpnH0KJ.png?alt=media&token=5b8e82d8-4502-4046-b0db-969f5cb3b78c

where a is the , v is the , and r is the radius of the circle.

Centripetal Force

The is defined as the force that is required to make an object move in a circular path. It is equal to the mass of the object multiplied by the . The formula for is: F = m * a_c, where F is the , m is the mass of the object and a_c is the .

Recall that the is not a new force; it is just another name for the directed toward the center of the circle. This could be caused by the normal force, tension, , , or another type of force.

For example, when a car takes a turn, the tires exert on the road, and the force is what provides the to keep the car moving in a circular path. Similarly, when a planet is orbiting the sun, it is the of the sun that acts as the .

The direction of the vector is always pointed toward the center of the circle, like the vector. This occurs because the accleration vector always points in the same direction as the vector (in this case, the ) due to Newton's Second Law (F=ma).

It's also important to note that the greater the speed of an object, the greater the required to keep it moving in a circular path. Similarly, the smaller the radius of a circular path, the greater the required to keep an object moving in that path.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%2011.58.44%20AM.png?alt=media&token=10083a60-4ccf-4f13-b6ab-41fa1338a205

Image courtesy of quickmeme.com.

Practice Questions

1. What is the direction of the of an object undergoing ? A) Radial B) Tangential C) Perpendicular D) Horizontal

Answer: A) Radial

2. An object of mass 5 kg is moving in a circular path of radius 3 m with a constant of 4 m/s. What is the of the object? A) 12 m/s^2 B) 16 m/s^2 C) 20 m/s^2 D) 5.33 m/s^2

Answer: D) 5.33 m/s^2 Explanation: To find the , we use the formula a_c = v^2/r, where a_c is the , v is the of the object and r is the radius of the circular path. Substituting the given values, we get a_c = (4 m/s)^2 / 3 m = 16 m/s^2 / 3 m = 5.33 m/s^2

3. A ball of mass 1 kg is tied to a string and is moving in a circular path of radius 0.5 m. If the ball is moving with a of 3 m/s, what is the acting on the ball? A) 2.25 N B) 4.5 N C) 9 N D) 1.5 N

Answer: D) 1.5 N Explanation: The acting on an object in is given by the formula F = m * a_c, where F is the , m is the mass of the object and a_c is the . To find the , we use the formula a_c = v^2/r, where a_c is the , v is the of the object and r is the radius of the circular path. By substituting the given values, we get a_c = (3 m/s)^2 / 0.5 m = 9 m^2/s^2 / 0.5 m = 18 m/s^2Then we can use the formula F = m * a_c to find the acting on the ball, we know the mass of the ball is 1 kg, so we substitute the values into the formula: F = m * a_c = 1 kg * 18 m/s^2 = 18 N

Key Terms to Review (17)

Acceleration

: Acceleration refers to the rate at which an object's velocity changes over time. It can be positive (speeding up), negative (slowing down), or zero (constant speed).

Center of Mass (COM)

: Center of mass refers to the point where all mass can be considered concentrated for purposes of calculating motion. It represents the average position of all particles comprising an object.

Center of Mass Velocity

: Center of mass velocity refers to the velocity of an object's center of mass, which represents its average motion.

Centripetal Acceleration

: Centripetal acceleration is the acceleration experienced by an object moving in a circular path. It is always directed towards the center of the circle and its magnitude can be calculated using the formula a = v^2/r, where v is the velocity of the object and r is the radius of the circle.

Centripetal Force

: Centripetal force is the net force acting towards the center of an object's circular path. It keeps objects moving in uniform circular motion by continuously changing their direction but not their speed.

Friction

: Friction is a force that opposes relative motion between two surfaces in contact. It arises due to microscopic irregularities between surfaces and can cause objects to slow down or come to rest.

Gravitational Force

: Gravitational force is the attractive force between two objects with mass. It depends on the masses of the objects and the distance between them.

Net Force

: The net force is the overall force acting on an object, taking into account both magnitude and direction. It determines the object's acceleration or deceleration.

Newton's Second Law (F=ma)

: Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In simpler terms, it explains how the motion of an object changes when a force is applied to it.

Pivot Point

: The pivot point is the fixed point around which an object rotates or turns.

Radial Acceleration

: Radial acceleration refers specifically to acceleration directed towards or away from the center of rotation. It occurs when there is either speeding up or slowing down while moving along a curved path.

Resultant Vector

: The resultant vector is the sum of two or more vectors. It represents the combined effect of all individual vectors.

Tangent

: In physics, tangent refers to a line that touches but does not cross or intersect with another curve at one specific point.

Uniform Circular Motion

: Uniform circular motion refers to an object moving along a circular path with constant speed. The object experiences acceleration towards the center of the circle, known as centripetal acceleration.

Vector Addition

: Vector addition is the process of combining two or more vectors to find their resultant vector. It involves adding the magnitudes and directions of the vectors.

Vector Quantity

: A vector quantity is a physical quantity that has both magnitude and direction. It can be represented by an arrow, where the length of the arrow represents the magnitude and the direction of the arrow represents the direction.

Velocity

: Velocity refers to the rate at which an object changes its position in a specific direction. It includes both speed and direction.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.