Exam #2 Study Guide: Newton's Laws, Forces, Work, Energy, and Conservation

Exam #2 Study Guide: Newton's Laws, Forces, Work, Energy, and Conservation

Overview

This study guide covers the main topics and concepts for Exam #2, focusing on Chapters 5 through 8. The exam will include conceptual questions, short numerical problems, and longer multi-step problems. The following notes summarize the key concepts, definitions, equations, and applications relevant to the exam.

Chapter 5: Newton's Laws of Motion

Newton's Three Laws of Motion

• First Law (Law of Inertia): An object at rest remains at rest, and an object in motion continues in motion with constant velocity unless acted upon by a net external force.

• Second Law: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

• Third Law: For every action, there is an equal and opposite reaction.

Equation (Second Law):

$F_{\text{net}} = m a$

Weight

• Definition: The gravitational force exerted on an object by the Earth.

• Equation:

$w = m g$

• where g is the acceleration due to gravity (approximately 9.8 m/s2 on Earth).

Normal Force

• Definition: The perpendicular contact force exerted by a surface on an object resting on it.

• For an object at rest on a horizontal surface with no other vertical forces:

$F_N = m g$

• More generally, the normal force balances the sum of all vertical forces.

Chapter 6: Applications of Newton's Laws

Friction Forces

• Static Friction: The force that prevents the initiation of sliding motion between two surfaces.

• Kinetic Friction: The force that opposes the motion of two surfaces sliding past each other.

• Equations:

$f_s \leq \mu_s F_N$

$f_k = \mu_k F_N$

• where \mu_s and \mu_k are the coefficients of static and kinetic friction, respectively.

Tension Force and Pulleys

• Tension: The pulling force transmitted through a string, rope, or cable.

• In ideal pulleys (massless and frictionless), tension is the same throughout the rope.

Elastic Force

• Hooke's Law: The force exerted by a spring is proportional to its displacement from equilibrium.

• Equation:

$F_{\text{spring}} = -k x$

• where k is the spring constant and x is the displacement from equilibrium.

Equilibrium and Non-Equilibrium Force Scenarios

• Equilibrium: The net force on an object is zero; the object is at rest or moves with constant velocity.

• Non-Equilibrium: The net force is nonzero; the object accelerates according to Newton's Second Law.

Centripetal Acceleration

• Definition: The acceleration directed toward the center of a circular path, keeping an object in circular motion.

• Equation:

$a_c = \frac{v^2}{r}$

• where v is the tangential speed and r is the radius of the circle.

Chapter 7: Work and Kinetic Energy

Work

• Definition: Work is done when a force causes displacement in the direction of the force.

• Equation:

$W = F d \cos \theta$

• where F is the force, d is the displacement, and \theta is the angle between force and displacement.

Kinetic Energy

• Definition: The energy of motion.

• Equation:

$KE = \frac{1}{2} m v^2$

Work-Energy Theorem

• The net work done on an object equals the change in its kinetic energy.

$W_{\text{net}} = \Delta KE$

Power

• Definition: The rate at which work is done.

• Equation:

$P = \frac{W}{t}$

• where W is work and t is time.

Force vs. Position Graphs

• The area under a force vs. position graph represents the work done by the force over a displacement.

Chapter 8: Potential Energy and Conservation of Energy

Gravitational Potential Energy

• Definition: Energy stored due to an object's position in a gravitational field.

• Equation:

$PE_g = m g h$

• where h is the height above a reference point.

Elastic Potential Energy

• Definition: Energy stored in a stretched or compressed spring.

• Equation:

$PE_{\text{elastic}} = \frac{1}{2} k x^2$

Conservation of Energy

• Principle: The total mechanical energy (kinetic + potential) of a system remains constant if only conservative forces act.

• Equation:

$KE_i + PE_i = KE_f + PE_f$

Non-Conservative Forces and Energy

• Non-conservative forces (e.g., friction) cause mechanical energy to be transformed into other forms (e.g., heat).

• Work by Non-Conservative Forces:

$W_{\text{nc}} = \Delta KE + \Delta PE$

Exam Structure and Question Types

Question Types

• Conceptual Questions: Multiple choice and one short answer; focus on understanding principles.

• Short Numerical Problems: Require selecting and applying the correct equation; typically single-step calculations.

• Longer Numerical Problems: Multi-step problems involving deeper analysis and synthesis of concepts.

Question Breakdown

Type	Percentage of Exam	Topics
Multiple Choice & Short Answer	18%	1-2 questions per chapter
Short Numerical Problems	47%	Weight, normal force, equilibrium, centripetal acceleration, work-energy theorem, force vs. position graph, power, energy conservation with springs
Longer Numerical Problems	35%	2-object force problems with acceleration, energy problems with non-conservative work

Key Equations Summary

Concept	Equation
Newton's Second Law	$F_{\text{net}} = m a$
Weight	$w = m g$
Friction (Static/Kinetic)	$f_s \leq \mu_s F_N$, $f_k = \mu_k F_N$
Hooke's Law	$F_{\text{spring}} = -k x$
Centripetal Acceleration	$a_c = \frac{v^2}{r}$
Work	$W = F d \cos \theta$
Kinetic Energy	$KE = \frac{1}{2} m v^2$
Power	$P = \frac{W}{t}$
Gravitational Potential Energy	$PE_g = m g h$
Elastic Potential Energy	$PE_{\text{elastic}} = \frac{1}{2} k x^2$
Conservation of Energy	$KE_i + PE_i = KE_f + PE_f$
Work by Non-Conservative Forces	$W_{\text{nc}} = \Delta KE + \Delta PE$

Example Applications

• Calculating the normal force: For a 10 kg box on a horizontal surface, $F_N = m g = 10 \times 9.8 = 98$ N.

• Finding work done by a force: A 5 N force moves an object 3 m at 60° to the direction of motion: $W = 5 \times 3 \times \cos 60^\circ = 7.5$ J.

• Energy conservation with a spring: A mass compresses a spring and is released; use $\frac{1}{2} k x^2 = \frac{1}{2} m v^2$ to find the speed at equilibrium.

Study Tips

• Review all key equations and understand when to apply each.

• Practice drawing free-body diagrams for force problems.

• Work through sample problems involving energy conservation, especially with springs and non-conservative forces.

• Be prepared for both conceptual and multi-step calculation questions.

Additional info: This guide expands on the exam outline by providing definitions, equations, and example applications for each topic. Students should refer to their textbook and class notes for more detailed derivations and problem-solving strategies.