Physics I: Dynamics, Energy, and Collisions – Exam Study Guide

Useful Information and Key Equations

Newton's Second Law

Newton's Second Law relates the net force acting on an object to its mass and acceleration.

• Equation: $F_{net} = \sum F_i = m a$

• Application: Used to analyze forces in linear and rotational motion.

Centripetal Acceleration

Centripetal acceleration is the acceleration experienced by an object moving in a circle at constant speed.

• Equation: $a_c = \frac{v^2}{r} = \omega^2 r$

• Application: Relevant for objects in circular motion, such as the sphere in question 1.

Momentum

Momentum is a measure of the motion of an object and is the product of its mass and velocity.

• Equation: $p = m v$

• Conservation: In the absence of external forces, total momentum is conserved.

Impulse

Impulse is the change in momentum resulting from a force applied over a time interval.

• Equation: $J = \Delta p = \int_{t_1}^{t_2} F(t) dt$

• Application: Used to analyze collisions and force interactions over time.

Energy

Energy is the capacity to do work. Several forms are relevant in mechanics:

• Kinetic Energy: $\frac{1}{2} m v^2$

• Gravitational Potential Energy: $m g y$

• Elastic Potential Energy: $\frac{1}{2} k (\Delta s)^2$

Elastic Collisions (with mass 2 initially at rest)

Elastic collisions conserve both momentum and kinetic energy. For two masses, where mass 2 is initially at rest:

• Final velocity of mass 1: $v_{f1} = \frac{m_1 - m_2}{m_1 + m_2} v_{i1}$

• Final velocity of mass 2: $v_{f2} = \frac{2 m_1}{m_1 + m_2} v_{i1}$

Work

Work is the energy transferred by a force acting over a distance.

• General Equation: $W = \int_{x_1}^{x_2} F_x(x) dx$

• Constant Force: $W = F_x \Delta x$

Work-Energy Theorem

The work-energy theorem relates the net work done on an object to its change in kinetic energy.

• Equation: $\Delta K = W_{net}$

• Alternate Form: $\Delta E_{mech} = W_{nc}$ (where $W_{nc}$ is work done by non-conservative forces)

Exam Questions and Concepts

1. Tension in Wires for Circular Motion

This problem involves a sphere tied to two wires, revolving in a horizontal circle. The goal is to find the tension in each wire.

• Key Concepts:

  • Centripetal force required for circular motion: $F_c = m \frac{v^2}{r}$

  • Decomposition of tension forces into components

  • Equilibrium in vertical and horizontal directions

• Example: A 0.50 kg sphere moving at 8.0 m/s in a circle, with wires at 30° angles, requires calculation of tension using force balance and trigonometry.

2. Pulley System and Friction

This question involves a block sliding on a table, pulled by a hanging mass via a pulley, with friction present.

• Key Concepts:

  • Newton's Second Law for each block

  • Relationship between accelerations due to pulley constraints

  • Kinetic friction: $f_k = \mu_k N$

• Example: Block A (7.0 kg) is pulled by Block B (7.0 kg) with $\mu_k = 0.20$. The acceleration of A is twice that of B due to the pulley arrangement.

3. Elastic Collision and Pendulum Rise

This problem involves a block colliding elastically with a pendulum bob, and asks how high the bob rises after the collision.

• Key Concepts:

  • Conservation of momentum and kinetic energy in elastic collisions

  • Conversion of kinetic energy to gravitational potential energy

• Example: A 0.25 kg block slides at 1.25 m/s and collides with a 0.75 kg pendulum bob. After the collision, the bob swings upward, converting its kinetic energy to potential energy: $m g h = \frac{1}{2} m v^2$.

4. Friction Force on a Hockey Puck

This question asks for the magnitude of the friction force experienced by a puck as it passes through a sticky spot on a frictionless surface.

• Key Concepts:

  • Impulse-momentum theorem: $F \Delta t = m \Delta v$

  • Work-energy theorem: $W = \Delta K$

• Example: A 200 g puck slows from 3.3 m/s to 1.4 m/s over 0.20 m. The friction force can be found using $W = F d = \Delta K$.

Summary Table: Key Equations and Their Applications

Equation	Physical Principle	Application Example
$F_{net} = m a$	Newton's Second Law	Calculating acceleration of blocks and tension in wires
$a_c = \frac{v^2}{r}$	Centripetal Acceleration	Sphere in circular motion (Question 1)
$p = m v$	Momentum	Elastic collision between block and pendulum bob (Question 3)
$K = \frac{1}{2} m v^2$	Kinetic Energy	Work-energy calculations for puck and pendulum
$W = F_x \Delta x$	Work	Friction force on hockey puck (Question 4)
$f_k = \mu_k N$	Kinetic Friction	Block sliding on table (Question 2)

Additional info:

• All questions are typical of introductory college physics, focusing on Newtonian mechanics, energy, and collisions.

• Diagrams and hints provided in the questions help clarify the physical setup and constraints.

• Students should be comfortable with vector decomposition, free-body diagrams, and applying conservation laws.