Physics 111 Exam 3 Practice Problems: Rotation, Dynamics, and Gravitation

Rotation of Rigid Bodies

Angular Kinematics and Dynamics

Rotational motion involves the movement of objects around a fixed axis. Key quantities include angular velocity, angular acceleration, and moment of inertia.

• Angular Displacement (θ): The angle through which a point or line has been rotated in a specified sense about a specified axis.

• Angular Velocity (ω): The rate of change of angular displacement, measured in radians per second (rad/s).

• Angular Acceleration (α): The rate of change of angular velocity, measured in rad/s2.

• Moment of Inertia (I): A measure of an object's resistance to changes in its rotation, dependent on mass distribution.

Key Equations:

• $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$

• $\omega = \omega_0 + \alpha t$

• $\omega^2 = \omega_0^2 + 2\alpha (\theta - \theta_0)$

• $I = \sum m_i r_i^2$ (for discrete masses)

Example: A wheel with radius 0.45 m slows from 2.5 rad/s to rest with angular acceleration -2 rad/s2. Find the distance a point on the rim travels before stopping.

Dynamics of Rotational Motion

Torque and Rotational Work

Torque is the rotational equivalent of force, causing angular acceleration. The net torque on a body is related to its moment of inertia and angular acceleration.

• Torque (τ): $\tau = r F \sin\theta$

• Newton's Second Law for Rotation: $\tau_{net} = I \alpha$

• Rotational Work: $W = \tau \theta$

Example: A force acts on a machine part at a given position; calculate the torque and verify its direction.

Conservation Laws in Rotational Motion

Conservation of Angular Momentum

Angular momentum is conserved in a system with no external torques. This principle is crucial in analyzing collisions and rotational dynamics.

• Angular Momentum (L): $L = I \omega$

• Conservation Law: $L_{initial} = L_{final}$ (if $\tau_{ext} = 0$)

Example: A merry-go-round problem where a child moves from the center to the rim, changing the system's angular velocity.

Rolling Motion and Energy

Rolling Without Slipping

When an object rolls without slipping, its rotational and translational motions are related. The point of contact with the surface is momentarily at rest.

• Condition for Rolling Without Slipping: $v_{cm} = R \omega$

• Kinetic Energy of Rolling Object: $K = \frac{1}{2} M v_{cm}^2 + \frac{1}{2} I \omega^2$

Example: A solid ball or a hollow shell rolling down an incline; find acceleration, friction, and energy at the bottom.

Collisions and Impulse in Rotational Systems

Impulse and Change in Momentum

Impulse is the product of force and the time interval over which it acts, resulting in a change in momentum. In rotational systems, angular impulse changes angular momentum.

• Impulse (J): $J = F \Delta t$

• Change in Momentum: $\Delta p = J$

• Angular Impulse: $\tau \Delta t = \Delta L$

Example: A ball rebounds off a wall; calculate the impulse and average force during contact.

Gravitation

Newton's Law of Universal Gravitation

Gravitational force is the attractive force between two masses, proportional to the product of their masses and inversely proportional to the square of the distance between them.

• Gravitational Force: $F = G \frac{m_1 m_2}{r^2}$

• Gravitational Constant: $G = 6.674 \times 10^{-11} \text{N} \cdot \text{m}^2/\text{kg}^2$

Example: Calculate the center of mass of the Sun-Earth system and the gravitational force between them.

Center of Mass

Finding the Center of Mass

The center of mass is the point at which the mass of a system or object can be considered to be concentrated for translational motion analysis.

• Center of Mass (x-coordinate): $x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$

• Center of Mass (y-coordinate): $y_{cm} = \frac{\sum m_i y_i}{\sum m_i}$

Example: Find the center of mass for a system of three particles at given coordinates.

Tabular Summary: Rotational Inertia of Common Objects

Object	Moment of Inertia (I)
Solid Sphere	$\frac{2}{5}MR^2$
Hollow Sphere	$\frac{2}{3}MR^2$
Solid Cylinder	$\frac{1}{2}MR^2$
Thin Rod (center)	$\frac{1}{12}ML^2$
Thin Rod (end)	$\frac{1}{3}ML^2$

Additional info: These formulas are essential for solving problems involving rolling, rotation, and energy calculations.

Sample Problem Types Covered

• Impulse and momentum change in collisions (linear and rotational)

• Rolling motion on inclines and horizontal surfaces

• Rotational kinematics and dynamics (angular acceleration, velocity, displacement)

• Conservation of angular momentum in systems with changing mass distribution

• Calculation of center of mass for particle systems

• Gravitational force and center of mass in two-body systems

Example Application: A bullet embeds in a block; use conservation of momentum to find the bullet's speed.