5 min readβ’january 15, 2023

Peter Apps

Kashvi Panjolia

Up until this point in the course, all of the motion covered has been described in terms of linear terms (forces, velocities, displacements, etc.) However a great deal of motion in the world isnβt about an object traveling anywhere, but instead **rotating **around a fixed axis (wheels, Merry-Go-Round, record/CD/DVD players).

This unit looks at the concepts already covered in units 1-6 and applies them to a rotating object instead of an object moving in a straight line. These topics will account for ~10-16% of the AP exam questions and will take approximately 12-17 45 minute class periods to cover.

**Big Idea #3: Force Interactions**- The interactions of an object with other objects can be described by forces.**Big Idea #4: Change**- Interactions between systems can result in changes in those systems.**Big Idea #5: Conservation**- Changes that occur as a result of interactions are constrained by conservation laws.

- Angular Displacement (π)
- Angular Velocity (π)
- Angular Acceleration (πΌ)
- Period (
*T*) - Torque (π)
- Moment of Inertia (
*I*) - Rotational Kinetic Energy (
*Krot*) - Angular Momentum (
*L)*

Rotational kinematics also involves the use of equations of motion for rotational motion. There are three **rotational kinematics equations** that are analogous to linear kinematics:

where Οf is the final angular velocity, Οi is the initial angular velocity, Ξ± is the angular acceleration, Ξt is the time, and ΞΞΈ is the angular displacement.

It's important to note that angular displacement, velocity, and acceleration are all vector quantities, meaning they have both magnitude and direction. In physics, **counterclockwise** rotation is considered positive, while **clockwise** rotation is considered negative. This is not what you would expect, so make sure to practice identifying positive and negative directions so you can nail this concept. π

π = rFsinΞΈ

where ΞΈ is the angle made by the lever arm vector (r) and the force vector (F) when the two vectors are placed **tail-to-tail**. βοΈβοΈ

The relationship between torque and angular acceleration can be represented by Newton's Second Law for rotation, which states that the net torque acting on an object is equal to the product of the object's moment of inertia (I) and its angular acceleration (Ξ±).

Ξ£π = IΞ±

The **moment of inertia (I)** is a measure of an object's resistance to rotational motion. It is measured in units of kilogram-meter squared (**kgm^2**). It depends on the object's mass and how far away that mass is from the axis of rotation. Inertia is a **scalar** quantity, while torque is a vector quantity. The moment of inertia of an object can be represented as:
I = cMR^2

where M is the mass of the object, R is the radius of the object, and c is a constant determined by the shape you are working with.

This relationship allows us to calculate the angular acceleration of an object given the net torque acting on it and its moment of inertia.

The angular momentum of an object can be calculated using the following equation:

where L is the angular momentum, I is the moment of inertia of the object and Ο is the angular velocity of the object.

There are two more equations for angular momentum that can be used to analyze the angular motion of an object:

1. The angular momentum equation for constant angular acceleration: **ΞL = πΞt**

This equation relates the change in angular momentum (ΞL) to the net torque (Ο) acting on it over a period of time (Ξt).

2. The angular momentum equation relative to a fixed point for an object moving in a straight line: **L=mvr**

In this equation, m is the mass of the object, measured in kilograms (kg), v is the linear velocity of a point on the object, measured in meters per second (m/s), and r is the distance from the axis of rotation to the point on the object, measured in meters (m).

These equations can be used to analyze the angular motion of an object and to understand how torque, angular velocity, and angular momentum are related.

Conservation of angular momentum states that the total angular momentum of a **closed** system (a system where no net external torque is acting on it) remains constant over time. This means that if the net torque acting on an object or system is zero, the angular momentum of the object or system will not change.

For example, if a figure skater is spinning with her arms extended and then brings her arms in close to her body, the angular momentum of her body will remain the same. The skater's moment of inertia will decrease as she brings her arms in closer to her body, but her angular velocity will increase by the same proportion to compensate and keep the angular momentum constant.

π₯**Watch: AP Physics 1 - ****Unit 7 Streams**

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π‘Unit 7 β Torque & Rotational Motion

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