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3.8 Applications of Circular Motion and Gravitation

8 min readjanuary 4, 2023

Peter Apps

Peter Apps

Daniella Garcia-Loos

Daniella Garcia-Loos

Kashvi Panjolia

Kashvi Panjolia

Peter Apps

Peter Apps

Daniella Garcia-Loos

Daniella Garcia-Loos

Kashvi Panjolia

Kashvi Panjolia

Frames of Reference

An observer in a reference frame can describe the motion of an object using such quantities as position, displacement, distance, velocity, speed, and acceleration. Depending on the , the direction and magnitude of these quantities will be different. A person watching a train from a side view will think the train is moving faster (has a greater velocity) than a person watching the train from behind the train.

Key Concept: Frame of Reference

A coordinate system in relation to which judgments can be made, usually from an observer’s point of view, is known as a . A moving with constant velocity is known as an inertial . 🙋‍♀️🙋‍♂️

Here are some key points about inertial frames of reference:

  • An inertial is a in which Newton's laws of motion hold true.

  • In an inertial , objects remain at rest or in motion at a constant velocity unless acted upon by a force.

  • Inertial frames of reference are also called Galilean frames of reference, named after the scientist Galileo Galilei who first described them.

  • In contrast, a non-inertial is a in which Newton's laws do not hold true. This can happen when the is accelerating or when there are strong gravitational influences present.

Real-world examples of inertial frames of reference:

  • A person watching a train go by while standing still at a train station

  • A stationary spaceship in deep space, far from any gravitational influences

  • A car traveling at a constant speed on a straight road

  • An airplane flying at a constant altitude and speed

Rotational Velocity and Acceleration:

Here are some key points about rotational velocity and acceleration:

  • Rotational velocity (aka ) is the speed at which an object is rotating. It is usually measured in units of radians per second (rad/s). It is denoted by the Greek letter omega ω and is the rotational analog to linear velocity.

  • Rotational acceleration (aka ) is the rate at which an object's rotational velocity changes. It is usually measured in units of radians per second squared (rad/s^2). It is denoted by the Greek letter alpha α and is the rotational analog to linear acceleration.

  • The rotational analog to linear position is the angle the object is rotating through, measured in radians (rad). Make sure you convert to radians if you are measuring the angle in degrees. The angle is denoted by the Greek letter theta θ.

  • Rotational velocity and acceleration are related to linear velocity and acceleration through the concept of and acceleration. is the velocity of a point on an object as it rotates, and is the acceleration of a point on an object as it rotates.

  • The relationship between rotational and and acceleration can be expressed using the formulas:

v = ωr ( = rotational velocity * radius)

a = αr ( = rotational acceleration * radius)

  • Rotational velocity and acceleration can be calculated using the formulas:

ω = (θf - θi)/(tf - ti) <-- Rotational velocity = (final angle - initial angle)/(final time - initial time)

ω = Δθ/Δt <--Rotational velocity = change in angle/change in time

α = Δω/Δt <-- Rotational acceleration = change in rotational velocity/change in time

Rotational Kinematics:

Each of the kinematics equations has a rotational analog.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202023-01-04%20at%205.46-S7twIE71Grgn.png?alt=media&token=88d92f15-2075-43b0-be34-ad7a76ddf9ae

These rotational analogues are true because Newton's laws also apply to rotational motion, allowing the derivation of these equations. Like how the linear kinematics equations are only true when the acceleration is constant, the rotational kinematics equations are only true when the is constant. Circular motion, gravitation, and rotational motion are all examples of situations where you can use these rotational kinematics equations because the is constant. These equations are also applicable to situations where there is both a radial (centripetal) and .

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%2012.28.53%20PM.png?alt=media&token=94fcfd1d-2d65-49d1-9743-c66fad85a8ec

The two equations above are important to note. The first equation relates the velocity to the period, T, of the motion. The period of the motion is the number of seconds the object takes to complete one full circle in a circular path. Since we know that v = ωr, we can divide both sides by r to obtain the angular equivalent to the first equation : ω = 2π/T. This equation will help you determine the when you only have the period, and vice versa.

The second equation is the equation for the centripetal acceleration of an object in circular motion. The centripetal acceleration of an object in circular motion always points towards the center of the circle the object is making. For objects in orbit, they are making a circle around a planet, so the planet's gravitational force is the centripetal acceleration since it points towards the center of the planet, which is the center of the object's orbit. Now, we can use a = v^2/r when we analyze problems using Newton's Second Law to substitute in the acceleration in F = ma.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%2012.29.26%20PM.png?alt=media&token=c2b0a94e-85c7-4b92-8927-82fe199dbf8a

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Image courtesy of Giphy.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%2012.29.55%20PM.png?alt=media&token=837dab36-08a6-4916-a709-e26ca0d00682

🎥Watch: Khan Academy - Angular Velocity and Speed

Force Vectors

Forces are described by vectors.

Key Vocabulary: Vector

Quantities that are described by a size (magnitude) and a direction (ex. East, Up, Right, etc.) 

Example: The gas station is five miles west of the car

  • Force, Displacement, Velocity, and Acceleration are vector quantities

A force can be simply described as a push or pull. An object can be pushed or pulled in different directions and with different strengths. We know that a push or pull has both magnitude and direction (therefore, it is a vector quantity) and can vary considerably in both regards.

When multiple forces are exerted on an object, the vector sum of these forces is called the net force. The direction the net force vector points in will always be the direction the acceleration vector points in, although the magnitudes may not be the same. This is true because of Newton's Second Law: F = ma. Since m, the mass, is a scalar (a regular number that changes the length of the vector, not its direction), the force and acceleration vectors must point in the same direction for the equation to hold true.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%2012.40.22%20PM.png?alt=media&token=b7be55eb-d239-4342-819d-9f245f4c9cd6

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Force Interactions

A force exerted on an object is always due to the interaction of that object with another object. The is due to the pulling of the planet on an object. The is due to the pushing of the surface on the object.

Force is always the result of an interaction of two or more objects. No object has force on its own. Therefore, no object can exert a force on itself. When you clap your hands, one hand exerts a force on the other. When you throw a ball, it exerts a force on your hand and your hand exerts a force on it.

Newton's Third Law

If one object exerts a force on a second object, the second object always exerts a force of equal magnitude on the first object in the opposite direction.

Newton’s Third Law states, “For every action, there is an equal and opposite reaction.” Simply put, in every interaction, there is a pair of forces acting on the two interacting objects. The magnitude of the force on the first object equals the magnitude of the force on the second object. The direction of the force on the first object is the exact opposite of the direction of the force on the second object. Forces always come in pairs - equal and opposite action-reaction force pairs.

Note: The gravitational force on an object and the on an object are NOT a pair. Even though the and gravitational force are often exerted in equal and opposite directions when an object is on a flat surface, the same two objects do not interact. In the case of gravity, the planet exerts a downward force on the object, but in the case of the , the surface exerts an upward force on the object. These forces do not interact between the same two objects, and therefore are not a pair.

🎥Watch: AP Physics 1 - Unit 3 Streams

Example Problem:

Design an experiment to investigate the circular motion of an object that is attached to a string and swung in a circular path.

Solution:

To design an experiment to investigate the circular motion of an object that is attached to a string and swung in a circular path, we will need to consider the following factors:

  1. The object: The object of interest should be chosen based on the properties and characteristics that we want to study. For example, we might choose a ball, a disk, or a cylinder.

  2. The circular path: The object should be attached to a string and swung in a circular path. This can be achieved by using a pivot point, such as a hook or a stand.

  3. The : The of the object should be measured and recorded. This can be done using a stopwatch or a timer to measure the time it takes for the object to complete one full rotation.

  4. The radius: The radius of the circular path should be measured and recorded. This can be done by measuring the length of the string and the height of the pivot point.

  5. The mass: The mass of the object should be measured and recorded. This can be done using a scale or a balance.

  6. The acceleration: The acceleration of the object should be measured and recorded. This can be done using a spring scale or a force sensor.

  7. The data: The data collected during the experiment should be analyzed and graphed to investigate the relationship between the , radius, mass, and acceleration of the object.

Key Terms to Review (12)

Angular Acceleration

: Angular acceleration refers to the rate at which an object's angular velocity changes over time. It measures how quickly an object is speeding up or slowing down its rotation.

Angular Velocity

: Angular velocity refers to the rate at which an object rotates or moves in a circular path. It is measured in radians per second (rad/s).

Force Interactions

: Force interactions refer to the mutual action between two objects resulting from their interaction through forces. Forces always occur in pairs and act on different objects involved in an interaction.

Frame of Reference

: A frame of reference is a set of coordinates that are used to describe the position and motion of objects. It provides a point from which measurements can be made.

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.

Newton's Third Law

: Newton's Third Law states that for every action, there is an equal and opposite reaction. When one object exerts a force on another object, the second object exerts an equal but opposite force on the first.

Normal Force

: The normal force is the force exerted by a surface to support the weight of an object resting on it. It acts perpendicular to the surface.

Radians per Second Squared (rad/s^2)

: Radians per second squared is a unit used to measure angular acceleration. It represents the change in angular velocity per unit of time.

Scalar quantity

: A scalar quantity is a physical measurement that only has magnitude and no direction. It can be described by a single value.

Tangential Acceleration

: Tangential acceleration refers to the rate at which an object's tangential velocity changes. It measures how quickly an object is speeding up or slowing down along its circular path.

Tangential Velocity

: Tangential velocity refers to the instantaneous linear velocity of an object moving along a curved path. It represents how fast an object is moving tangent to its circular path at any given point.

Weight Force

: Weight force refers to the gravitational force acting on an object due to its mass and acceleration due to gravity.

3.8 Applications of Circular Motion and Gravitation

8 min readjanuary 4, 2023

Peter Apps

Peter Apps

Daniella Garcia-Loos

Daniella Garcia-Loos

Kashvi Panjolia

Kashvi Panjolia

Peter Apps

Peter Apps

Daniella Garcia-Loos

Daniella Garcia-Loos

Kashvi Panjolia

Kashvi Panjolia

Frames of Reference

An observer in a reference frame can describe the motion of an object using such quantities as position, displacement, distance, velocity, speed, and acceleration. Depending on the , the direction and magnitude of these quantities will be different. A person watching a train from a side view will think the train is moving faster (has a greater velocity) than a person watching the train from behind the train.

Key Concept: Frame of Reference

A coordinate system in relation to which judgments can be made, usually from an observer’s point of view, is known as a . A moving with constant velocity is known as an inertial . 🙋‍♀️🙋‍♂️

Here are some key points about inertial frames of reference:

  • An inertial is a in which Newton's laws of motion hold true.

  • In an inertial , objects remain at rest or in motion at a constant velocity unless acted upon by a force.

  • Inertial frames of reference are also called Galilean frames of reference, named after the scientist Galileo Galilei who first described them.

  • In contrast, a non-inertial is a in which Newton's laws do not hold true. This can happen when the is accelerating or when there are strong gravitational influences present.

Real-world examples of inertial frames of reference:

  • A person watching a train go by while standing still at a train station

  • A stationary spaceship in deep space, far from any gravitational influences

  • A car traveling at a constant speed on a straight road

  • An airplane flying at a constant altitude and speed

Rotational Velocity and Acceleration:

Here are some key points about rotational velocity and acceleration:

  • Rotational velocity (aka ) is the speed at which an object is rotating. It is usually measured in units of radians per second (rad/s). It is denoted by the Greek letter omega ω and is the rotational analog to linear velocity.

  • Rotational acceleration (aka ) is the rate at which an object's rotational velocity changes. It is usually measured in units of radians per second squared (rad/s^2). It is denoted by the Greek letter alpha α and is the rotational analog to linear acceleration.

  • The rotational analog to linear position is the angle the object is rotating through, measured in radians (rad). Make sure you convert to radians if you are measuring the angle in degrees. The angle is denoted by the Greek letter theta θ.

  • Rotational velocity and acceleration are related to linear velocity and acceleration through the concept of and acceleration. is the velocity of a point on an object as it rotates, and is the acceleration of a point on an object as it rotates.

  • The relationship between rotational and and acceleration can be expressed using the formulas:

v = ωr ( = rotational velocity * radius)

a = αr ( = rotational acceleration * radius)

  • Rotational velocity and acceleration can be calculated using the formulas:

ω = (θf - θi)/(tf - ti) <-- Rotational velocity = (final angle - initial angle)/(final time - initial time)

ω = Δθ/Δt <--Rotational velocity = change in angle/change in time

α = Δω/Δt <-- Rotational acceleration = change in rotational velocity/change in time

Rotational Kinematics:

Each of the kinematics equations has a rotational analog.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202023-01-04%20at%205.46-S7twIE71Grgn.png?alt=media&token=88d92f15-2075-43b0-be34-ad7a76ddf9ae

These rotational analogues are true because Newton's laws also apply to rotational motion, allowing the derivation of these equations. Like how the linear kinematics equations are only true when the acceleration is constant, the rotational kinematics equations are only true when the is constant. Circular motion, gravitation, and rotational motion are all examples of situations where you can use these rotational kinematics equations because the is constant. These equations are also applicable to situations where there is both a radial (centripetal) and .

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%2012.28.53%20PM.png?alt=media&token=94fcfd1d-2d65-49d1-9743-c66fad85a8ec

The two equations above are important to note. The first equation relates the velocity to the period, T, of the motion. The period of the motion is the number of seconds the object takes to complete one full circle in a circular path. Since we know that v = ωr, we can divide both sides by r to obtain the angular equivalent to the first equation : ω = 2π/T. This equation will help you determine the when you only have the period, and vice versa.

The second equation is the equation for the centripetal acceleration of an object in circular motion. The centripetal acceleration of an object in circular motion always points towards the center of the circle the object is making. For objects in orbit, they are making a circle around a planet, so the planet's gravitational force is the centripetal acceleration since it points towards the center of the planet, which is the center of the object's orbit. Now, we can use a = v^2/r when we analyze problems using Newton's Second Law to substitute in the acceleration in F = ma.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%2012.29.26%20PM.png?alt=media&token=c2b0a94e-85c7-4b92-8927-82fe199dbf8a

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fcyclist.gif?alt=media&token=ee764a00-a7ee-458a-89d1-e1ff85eb842b

Image courtesy of Giphy.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%2012.29.55%20PM.png?alt=media&token=837dab36-08a6-4916-a709-e26ca0d00682

🎥Watch: Khan Academy - Angular Velocity and Speed

Force Vectors

Forces are described by vectors.

Key Vocabulary: Vector

Quantities that are described by a size (magnitude) and a direction (ex. East, Up, Right, etc.) 

Example: The gas station is five miles west of the car

  • Force, Displacement, Velocity, and Acceleration are vector quantities

A force can be simply described as a push or pull. An object can be pushed or pulled in different directions and with different strengths. We know that a push or pull has both magnitude and direction (therefore, it is a vector quantity) and can vary considerably in both regards.

When multiple forces are exerted on an object, the vector sum of these forces is called the net force. The direction the net force vector points in will always be the direction the acceleration vector points in, although the magnitudes may not be the same. This is true because of Newton's Second Law: F = ma. Since m, the mass, is a scalar (a regular number that changes the length of the vector, not its direction), the force and acceleration vectors must point in the same direction for the equation to hold true.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%2012.40.22%20PM.png?alt=media&token=b7be55eb-d239-4342-819d-9f245f4c9cd6

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-06-03%20at%2012.51-l6FX8qAklY4y.png?alt=media&token=bd8e5f5c-2e9c-4d98-98c6-bd52244ec0e5

Force Interactions

A force exerted on an object is always due to the interaction of that object with another object. The is due to the pulling of the planet on an object. The is due to the pushing of the surface on the object.

Force is always the result of an interaction of two or more objects. No object has force on its own. Therefore, no object can exert a force on itself. When you clap your hands, one hand exerts a force on the other. When you throw a ball, it exerts a force on your hand and your hand exerts a force on it.

Newton's Third Law

If one object exerts a force on a second object, the second object always exerts a force of equal magnitude on the first object in the opposite direction.

Newton’s Third Law states, “For every action, there is an equal and opposite reaction.” Simply put, in every interaction, there is a pair of forces acting on the two interacting objects. The magnitude of the force on the first object equals the magnitude of the force on the second object. The direction of the force on the first object is the exact opposite of the direction of the force on the second object. Forces always come in pairs - equal and opposite action-reaction force pairs.

Note: The gravitational force on an object and the on an object are NOT a pair. Even though the and gravitational force are often exerted in equal and opposite directions when an object is on a flat surface, the same two objects do not interact. In the case of gravity, the planet exerts a downward force on the object, but in the case of the , the surface exerts an upward force on the object. These forces do not interact between the same two objects, and therefore are not a pair.

🎥Watch: AP Physics 1 - Unit 3 Streams

Example Problem:

Design an experiment to investigate the circular motion of an object that is attached to a string and swung in a circular path.

Solution:

To design an experiment to investigate the circular motion of an object that is attached to a string and swung in a circular path, we will need to consider the following factors:

  1. The object: The object of interest should be chosen based on the properties and characteristics that we want to study. For example, we might choose a ball, a disk, or a cylinder.

  2. The circular path: The object should be attached to a string and swung in a circular path. This can be achieved by using a pivot point, such as a hook or a stand.

  3. The : The of the object should be measured and recorded. This can be done using a stopwatch or a timer to measure the time it takes for the object to complete one full rotation.

  4. The radius: The radius of the circular path should be measured and recorded. This can be done by measuring the length of the string and the height of the pivot point.

  5. The mass: The mass of the object should be measured and recorded. This can be done using a scale or a balance.

  6. The acceleration: The acceleration of the object should be measured and recorded. This can be done using a spring scale or a force sensor.

  7. The data: The data collected during the experiment should be analyzed and graphed to investigate the relationship between the , radius, mass, and acceleration of the object.

Key Terms to Review (12)

Angular Acceleration

: Angular acceleration refers to the rate at which an object's angular velocity changes over time. It measures how quickly an object is speeding up or slowing down its rotation.

Angular Velocity

: Angular velocity refers to the rate at which an object rotates or moves in a circular path. It is measured in radians per second (rad/s).

Force Interactions

: Force interactions refer to the mutual action between two objects resulting from their interaction through forces. Forces always occur in pairs and act on different objects involved in an interaction.

Frame of Reference

: A frame of reference is a set of coordinates that are used to describe the position and motion of objects. It provides a point from which measurements can be made.

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.

Newton's Third Law

: Newton's Third Law states that for every action, there is an equal and opposite reaction. When one object exerts a force on another object, the second object exerts an equal but opposite force on the first.

Normal Force

: The normal force is the force exerted by a surface to support the weight of an object resting on it. It acts perpendicular to the surface.

Radians per Second Squared (rad/s^2)

: Radians per second squared is a unit used to measure angular acceleration. It represents the change in angular velocity per unit of time.

Scalar quantity

: A scalar quantity is a physical measurement that only has magnitude and no direction. It can be described by a single value.

Tangential Acceleration

: Tangential acceleration refers to the rate at which an object's tangential velocity changes. It measures how quickly an object is speeding up or slowing down along its circular path.

Tangential Velocity

: Tangential velocity refers to the instantaneous linear velocity of an object moving along a curved path. It represents how fast an object is moving tangent to its circular path at any given point.

Weight Force

: Weight force refers to the gravitational force acting on an object due to its mass and acceleration due to gravity.


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© 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.