AP®︎ Physics 1: Unit 5 Practice Test

Essential Knowledge: 5.1.A.1

Angular displacement is the measurement of the angle, in radians, through which a point on a rigid system rotates about a specified axis. Relevant equation: $\Delta\theta = \theta - \theta_0$

Essential Knowledge: 5.1.A.1.i

A rigid system is one that holds its shape but in which different points on the system move in different directions during rotation. A rigid system cannot be modeled as an object.

Essential Knowledge: 5.1.A.1.ii

One direction of angular displacement about an axis of rotation—clockwise or counterclockwise—is typically indicated as mathematically positive, with the other direction becoming mathematically negative.

Essential Knowledge: 5.1.A.1.iii

If the rotation of a system about an axis may be well described using the motion of the system’s center of mass, the system may be treated as a single object. For example, the rotation of Earth about its axis may be considered negligible when considering the revolution of Earth about the center of mass of the Earth–Sun system.

Essential Knowledge: 5.1.A.2

Average angular velocity is the average rate at which angular position changes with respect to time. Relevant equation: $\omega_{\mathrm{avg}} = \frac{\Delta\theta}{\Delta t}$

Essential Knowledge: 5.1.A.3

Average angular acceleration is the average rate at which the angular velocity changes with respect to time. Relevant equation: $\alpha_{\mathrm{avg}} = \frac{\Delta\omega}{\Delta t}$

Essential Knowledge: 5.1.A.4

Angular displacement, angular velocity, and angular acceleration around one axis are analogous to linear displacement, velocity, and acceleration in one dimension and demonstrate the same mathematical relationships.

Essential Knowledge: 5.1.A.4.i

For constant angular acceleration, the mathematical relationships between angular displacement, angular velocity, and angular acceleration can be described with the following equations: $\omega = \omega_0 + \alpha t$; $\theta = \theta_0 + \omega_0t + \frac{1}{2}\alpha t^2$; $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

Essential Knowledge: 5.1.A.4.ii

Graphs of angular displacement, angular velocity, and angular acceleration as functions of time can be used to find the relationships between those quantities.

Essential Knowledge: 5.2.A.1

For a point at a distance $r$ from a fixed axis of rotation, the linear distance $s$ traveled by the point as the system rotates through an angle $\Delta\theta$ is given by the equation $\Delta s = r\Delta\theta$.

Essential Knowledge: 5.2.A.2

Derived relationships of linear velocity and of the tangential component of acceleration to their respective angular quantities are given by the following equations: $s = r\theta$; $v = r\omega$; $a_T = r\alpha$

Essential Knowledge: 5.2.A.3

For a rigid system, all points within that system have the same angular velocity and angular acceleration.

Essential Knowledge: 5.3.A.1

Torque results only from the force component perpendicular to the position vector from the axis of rotation to the point of application of the force.

Essential Knowledge: 5.3.A.2

The lever arm is the perpendicular distance from the axis of rotation to the line of action of the exerted force.

Essential Knowledge: 5.3.B.1

Torques can be described using force diagrams.

Essential Knowledge: 5.3.B.1.i

Force diagrams are similar to free-body diagrams and are used to analyze the torques exerted on a rigid system.

Essential Knowledge: 5.3.B.1.ii

Similar to free-body diagrams, force diagrams represent the relative magnitude and direction of the forces exerted on a rigid system. Force diagrams also depict the location at which those forces are exerted relative to the axis of rotation.

Essential Knowledge: 5.3.B.2

The magnitude of the torque exerted on a rigid system by a force is described by the following equation, where $\theta$ is the angle between the force vector and the position vector from the axis of rotation to the point of application of the force. $\tau = rF_{\perp} = rF\sin\theta$

Essential Knowledge: 5.4.A.1

Rotational inertia measures a rigid system’s resistance to changes in rotation and is related to the mass of the system and the distribution of that mass relative to the axis of rotation.

Essential Knowledge: 5.4.A.2

The rotational inertia of an object rotating a perpendicular distance $r$ from an axis is described by the equation $I = mr^2$.

Essential Knowledge: 5.4.A.3

The total rotational inertia of a collection of objects about an axis is the sum of the rotational inertias of each object about that axis: $I_{tot} = \sum I_i = \sum m_i r_i^2$

Essential Knowledge: 5.4.B.1

A rigid system’s rotational inertia in a given plane is at a minimum when the rotational axis passes through the system’s center of mass.

Essential Knowledge: 5.4.B.2

The parallel axis theorem uses the following equation to relate the rotational inertia of a rigid system about any axis that is parallel to an axis through its center of mass: $I' = I_{cm} + Md^2$

Essential Knowledge: 5.5.A.1

A system may exhibit rotational equilibrium (constant angular velocity) without being in translational equilibrium, and vice versa.

Essential Knowledge: 5.5.A.1.i

Free-body and force diagrams describe the nature of the forces and torques exerted on an object or rigid system.

Essential Knowledge: 5.5.A.1.ii

Rotational equilibrium is a configuration of torques such that the net torque exerted on the system is zero. Relevant equation: $\sum \tau_i = 0$

Essential Knowledge: 5.5.A.1.iii

The rotational analog of Newton’s first law is that a system will have a constant angular velocity only if the net torque exerted on the system is zero.

Essential Knowledge: 5.5.A.2

A rotational corollary to Newton’s second law states that if the torques exerted on a rigid system are not balanced, the system’s angular velocity must be changing.

Essential Knowledge: 5.6.A.1

Angular velocity changes when the net torque exerted on the object or system is not equal to zero.

Essential Knowledge: 5.6.A.2

The rate at which the angular velocity of a rigid system changes is directly proportional to the net torque exerted on the rigid system and is in the same direction. The angular acceleration of the rigid system is inversely proportional to the rotational inertia of the rigid system. Relevant equation: $\alpha_{\text{sys}} = \frac{\sum \tau}{I_{\text{sys}}} = \frac{\tau_{\text{net}}}{I_{\text{sys}}}$

Essential Knowledge: 5.6.A.3

To fully describe a rotating rigid system, linear and rotational analyses may need to be performed independently.