Physics Practice Exam Study Guide: Mechanics, Energy, Rotation, and Gravitation

Mechanics: Motion, Forces, and Energy

Simple Pendulum Motion

A simple pendulum consists of a mass attached to a string, swinging under the influence of gravity. The speed of the mass at its lowest point can be found using conservation of energy.

• Key Concept: Mechanical energy is conserved (potential energy converts to kinetic energy).

• Equation: $mgh = \frac{1}{2}mv^2$

• Example: A 0.205-kg mass on a 75-cm string released from rest horizontally; find speed at lowest point.

Spring Potential Energy and Spring Constant

Springs store energy when stretched or compressed. The spring constant quantifies the stiffness of the spring.

• Key Concept: Potential energy in a spring: $U = \frac{1}{2}kx^2$

• Equation: $k = \frac{2U}{x^2}$

• Example: A spring stores 18 J when stretched by 2.40 cm; find $k$.

Momentum and Collisions

Momentum is conserved in collisions. In a perfectly inelastic collision, objects stick together after impact.

• Key Concept: Conservation of momentum: $m_1v_1 + m_2v_2 = (m_1 + m_2)v_f$

• Example: Two objects collide and stick together; find final velocity.

Rotational Motion and Dynamics

Angular Acceleration and Velocity

Rotational motion involves angular velocity and acceleration. The angular acceleration is the rate of change of angular velocity.

• Key Concept: $\alpha = \frac{\Delta \omega}{\Delta t}$

• Example: A wheel with initial angular velocity, making several revolutions; find angular acceleration.

Rotational Kinetic Energy

Rotating objects possess kinetic energy due to their rotation, given by $K_{rot} = \frac{1}{2}I\omega^2$.

• Key Concept: $I$ is the moment of inertia, $\omega$ is angular velocity.

• Example: A sphere or cylinder rolling without slipping; calculate total kinetic energy.

Moment of Inertia

The moment of inertia quantifies an object's resistance to changes in rotational motion.

• Key Concept: For a solid sphere: $I = \frac{2}{5}MR^2$; for a thin rod about center: $I = \frac{1}{12}ML^2$

• Example: Calculate $I$ for a sphere or rod given mass and dimensions.

Work, Energy, and Conservation Laws

Conservation of Energy

Energy cannot be created or destroyed; it transforms between kinetic, potential, and other forms.

• Key Concept: $E_{initial} = E_{final}$

• Equation: $mgh + \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 = \text{constant}$

• Example: Block released from spring, ball rolling down incline, etc.

Impulse and Average Force

Impulse is the change in momentum, and average force can be found from impulse over time.

• Key Concept: $F_{avg} = \frac{\Delta p}{\Delta t}$

• Example: Tennis ball rebounding from wall; calculate average force.

Systems of Particles and Gravitation

Center of Mass

The center of mass is the average position of mass in a system, important for analyzing motion of composite objects.

• Key Concept: $x_{cm} = \frac{\sum m_ix_i}{\sum m_i}$

• Example: Sun-Earth system; find center of mass location.

Gravitational Forces

Newton's law of universal gravitation describes the attractive force between two masses.

• Key Concept: $F = G\frac{m_1m_2}{r^2}$

• Example: Calculate gravitational force between Earth and Sun.

Selected Formulas and Applications

Topic	Key Formula	Application
Spring Constant	$k = \frac{2U}{x^2}$	Find $k$ from energy and displacement
Conservation of Momentum	$m_1v_1 + m_2v_2 = (m_1 + m_2)v_f$	Final velocity after collision
Rotational Kinetic Energy	$K_{rot} = \frac{1}{2}I\omega^2$	Energy of rotating object
Moment of Inertia (Sphere)	$I = \frac{2}{5}MR^2$	Resistance to rotation
Impulse	$F_{avg} = \frac{\Delta p}{\Delta t}$	Average force during collision
Gravitational Force	$F = G\frac{m_1m_2}{r^2}$	Force between two masses

Additional info:

• Questions cover topics from Ch 02 (Motion Along a Straight Line), Ch 05 (Applying Newton's Laws), Ch 06 (Work & Kinetic Energy), Ch 07 (Potential Energy & Conservation), Ch 08 (Momentum, Impulse, and Collisions), Ch 09 (Rotation of Rigid Bodies), Ch 10 (Dynamics of Rotational Motion), Ch 13 (Gravitation), and Ch 11 (Equilibrium & Elasticity).

• Some questions involve diagrams (e.g., pulley systems, rolling objects) and require application of multiple concepts.

• Answers are provided on the last page for self-assessment.