Physics Practice Exam: Mechanics, Energy, and Rotational Motion

Mechanics 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 the lowest point can be found using conservation of energy.

• Key Concept: Mechanical energy is conserved (ignoring air resistance).

• Equation: $mgh = \frac{1}{2}mv^2$

• Application: The change in height determines the kinetic energy at the lowest point.

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

Spring Potential Energy

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: $k = \frac{2U}{x^2}$

• Application: Given energy and displacement, solve for spring constant.

• Example: 18 J stored when stretched by 2.40 cm; find $k$.

Conservation of Momentum in Collisions

When two objects collide and stick together, their combined velocity is found using conservation of momentum.

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

• Application: Used for perfectly inelastic collisions.

• Example: 2.3-kg and 3.5-kg objects collide; find final velocity.

Forces and Rotational Motion

Impulse and Average Force

Impulse is the product of force and time, and equals the change in momentum.

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

• Application: Used to find average force during a collision.

• Example: Tennis ball rebounds off wall; calculate average force.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity, often found from the number of revolutions and initial angular velocity.

• Key Concept: $\alpha = \frac{\omega_f - \omega_i}{t}$

• Application: Used when a wheel slows down after spinning.

• Example: Wheel with initial $\omega = 826$ rad/s makes 234 revolutions before stopping.

Maximum Speed in a Spring System

When a block attached to a spring is released from rest, its maximum speed occurs as it passes through equilibrium.

• Key Concept: Conservation of energy: $\frac{1}{2}kx^2 = \frac{1}{2}mv^2$

• Application: Used to find maximum speed after release.

• Example: 3.5-kg block, $k = 150$ N/m, displaced 20 cm.

Collisions and Conservation Laws

Bullet-Block Collision

When a bullet embeds in a block, conservation of momentum is used to find the bullet's speed.

• Key Concept: $m_{bullet}v_{bullet} = (m_{bullet} + m_{block})v_{final}$

• Application: Used for inelastic collisions.

• Example: 7.8-g bullet, 5.3-kg block, final speed 2.3 m/s.

Conservation of Energy on an Incline

Objects rolling up an incline convert kinetic energy to potential energy.

• Key Concept: $\frac{1}{2}mv^2 = mgh$

• Application: Used to find distance traveled up the incline.

• Example: Hoop with initial speed 10.0 m/s up 30° incline.

Angular Position and Acceleration

Angular position as a function of time can be differentiated to find angular velocity and acceleration.

• Key Concept: $\theta(t) = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$

• Application: Find $\alpha$ from given function.

• Example: $\theta = 4.0 - 6.0t + 2.0t^2$

Pulley Systems and Rotational Kinetics

Pulley and Block Systems

Two masses connected over a pulley accelerate due to gravity; use energy or force analysis to find speed.

• Key Concept: $m_1g - m_2g = (m_1 + m_2)a$

• Application: Used to find speed after a certain distance.

• Example: 6.0-kg and 3.0-kg blocks, heavier block descends 0.56 m.

Rolling Spheres and Conservation of Energy

Rolling objects down inclines convert potential energy to both translational and rotational kinetic energy.

• Key Concept: $mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$

• Application: Used for spheres, cylinders, and hoops.

• Example: Sphere of mass 0.13 kg, radius 13.3 cm, rolling down incline.

Rotational Inertia and Energy

Rotational Kinetic Energy

Rotational kinetic energy depends on the moment of inertia and angular velocity.

• Key Concept: $K_{rot} = \frac{1}{2}I\omega^2$

• Application: Used for rolling balls and spheres.

• Example: Bowling ball, $I = \frac{2}{5}MR^2$, speed 9.5 m/s.

Vertical Spring Oscillations

When a mass is attached to a vertical spring and released, it oscillates about the equilibrium position.

• Key Concept: $mg = kx$

• Application: Used to find maximum displacement.

• Example: 1.7-kg mass, $k = 121$ N/m.

Moment of Inertia

The moment of inertia quantifies an object's resistance to rotational acceleration.

• Key Concept: $I = \frac{2}{5}MR^2$ for a solid sphere, $I = MR^2$ for a hoop.

• Application: Used to calculate rotational energy and dynamics.

• Example: Sphere of mass 1.3 kg, radius 1.2 m.

Advanced Applications

Rocket Motion and Conservation of Momentum

When a rocket splits into two parts, conservation of momentum is used to find the velocity of each part.

• Key Concept: $m_1v_1 + m_2v_2 = m_{total}v_{initial}$

• Application: Used for explosions and separations.

• Example: Rocket of mass 1.0 kg splits; find velocity of one part.

Center of Mass in the Earth-Sun System

The center of mass of a two-body system is found using the masses and separation distance.

• Key Concept: $x_{CM} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2}$

• Application: Used for planetary systems.

• Example: Sun and Earth, given masses and distance.

Summary Table: Key Equations

Topic	Equation	Application
Conservation of Energy	$mgh = \frac{1}{2}mv^2$	Pendulum, rolling objects
Spring Potential Energy	$U = \frac{1}{2}kx^2$	Springs, oscillations
Conservation of Momentum	$m_1v_1 + m_2v_2 = (m_1 + m_2)v_f$	Collisions
Impulse	$F_{avg} = \frac{\Delta p}{\Delta t}$	Collisions, force calculations
Rotational Kinetic Energy	$K_{rot} = \frac{1}{2}I\omega^2$	Rotating objects
Moment of Inertia	$I = \frac{2}{5}MR^2$ (sphere), $I = MR^2$ (hoop)	Rotational motion
Spring Equilibrium	$mg = kx$	Vertical springs
Center of Mass	$x_{CM} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2}$	Two-body systems

Additional info: These questions cover topics from chapters on energy, momentum, rotational motion, springs, and center of mass, relevant to introductory college physics.