Physics Exam Review: Newton's Laws, Applications, Work, and Energy (Chapters 4–7)

Newton's Laws of Motion

Force

A force is a push or pull acting on an object, and it is the fundamental cause of acceleration. The net force is the vector sum of all forces acting on an object:

$\sum \vec{F}$

• If the net force is zero, the object's acceleration is zero:

$\sum \vec{F} = 0 \Rightarrow \vec{a} = 0$

• The object remains at rest or moves with constant velocity.

Newton's 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.

• This law explains that changes in motion require a net external force.

Mass and Inertia

• Mass is a measure of an object's resistance to changes in motion (inertia).

• Inertia is the tendency of an object to remain at rest or in uniform motion.

• Greater mass means greater inertia.

Mass vs. Weight

• Mass: Amount of matter in an object; independent of location.

• Weight: Gravitational force acting on an object; depends on location.

The weight of an object is:

$w = mg$

where $m$ is mass and $g$ is the acceleration due to gravity.

Newton's Second Law

• Relates net force to acceleration:

$\sum \vec{F} = m\vec{a}$

• Acceleration is in the direction of the net force.

• Larger net force produces larger acceleration; larger mass produces smaller acceleration for the same net force.

Newton's Third Law

• Forces always occur in pairs: If object A exerts a force on object B, then object B exerts an equal and opposite force on object A.

$\vec{F}_{AB} = -\vec{F}_{BA}$

• These forces act on different objects and do not cancel each other.

Applications of Newton's Laws

Normal Force

• The normal force is the support force exerted by a surface, acting perpendicular to the surface.

Friction

• Friction opposes motion or attempted motion between surfaces in contact.

Static Friction

• Prevents motion from starting:

$f_s \leq \mu_s N$

• Maximum static friction:

$f_{s,\text{max}} = \mu_s N$

Kinetic Friction

• Acts when the object is moving:

$f_k = \mu_k N$

• Usually, $\mu_s > \mu_k$ (harder to start moving than to keep moving).

• Friction always acts opposite to the direction of motion or tendency of motion.

Uniform Circular Motion

• Motion at constant speed in a circular path; velocity changes direction, so there is acceleration (centripetal acceleration):

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

• The net force causing this acceleration is the centripetal force:

$F_c = m\frac{v^2}{r}$

• Centripetal force always points toward the center of the circle.

Work and Kinetic Energy

Work

• Work is done when a force causes displacement:

$W = \vec{F} \cdot \vec{s}$

$W = Fs \cos \theta$

• Only the component of force parallel to displacement does work.

• If $\theta = 0^\circ$, work is maximum and positive; if $\theta = 90^\circ$, work is zero; if $\theta = 180^\circ$, work is negative.

• Positive work adds energy; negative work removes energy.

Example:

• Pushing a box forward: positive work.

• Friction acting opposite motion: negative work.

Kinetic Energy

• Kinetic energy is the energy of motion:

$K = \frac{1}{2}mv^2$

• Depends on both mass and speed (speed squared).

Work-Energy Theorem

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

$W_{\text{tot}} = \Delta K = K_f - K_i$

• If net work is positive, the object speeds up; if negative, it slows down; if zero, kinetic energy does not change.

• This theorem can simplify problem-solving by relating initial and final states directly.

Variable Forces

• If force varies with position, work is found by integration:

$W = \int_{x_i}^{x_f} F(x) dx$

• Especially important for springs and other non-constant forces.

Power

• Power is the rate of doing work or transferring energy:

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

• Instantaneous power (force and velocity in same direction):

$P = F_{||}v$

• More power means energy is transferred more quickly.

Potential Energy and Energy Conservation

Potential Energy

• Potential energy is stored energy due to position or configuration.

Gravitational Potential Energy

• Near Earth's surface:

$U_g = mgy$

• Raising an object increases its gravitational potential energy.

Elastic Potential Energy

• For a spring:

$U_s = \frac{1}{2}kx^2$

• $k$ is the spring constant; $x$ is displacement from equilibrium.

Conservative and Nonconservative Forces

• Conservative forces: Work depends only on initial and final positions (e.g., gravity, spring force).

• For conservative forces:

$W_c = -\Delta U$

• If a conservative force does positive work, potential energy decreases, and vice versa.

• Nonconservative forces: Work depends on the path (e.g., friction, air resistance); usually transform mechanical energy into other forms.

Mechanical Energy

• Total mechanical energy is the sum of kinetic and potential energy:

$E_{\text{mech}} = K + U$

Conservation of Mechanical Energy

• If only conservative forces do work, mechanical energy is conserved:

$K_i + U_i = K_f + U_f$

• Energy changes form but is not lost (e.g., falling object: $U_g$ decreases, $K$ increases).

Nonconservative Forces Present

• If nonconservative forces (like friction) act, mechanical energy is not conserved by itself:

$K_i + U_i + W_{nc} = K_f + U_f$

• Some mechanical energy is transformed into thermal or other forms.

When to Use Newton's Laws vs. Energy Methods

• Newton's laws: Useful when forces and accelerations are the main focus.

• Energy methods: Useful for comparing initial and final states, especially with complicated paths or when asked about speed, height, or stopping distance.

Key Equations Summary

Concept	Equation
Newton's Second Law	$\sum \vec{F} = m\vec{a}$
Equilibrium	$\sum \vec{F} = 0$
Weight	$w = mg$
Newton's Third Law	$\vec{F}_{AB} = -\vec{F}_{BA}$
Static Friction	$f_s \leq \mu_s N$
Kinetic Friction	$f_k = \mu_k N$
Centripetal Acceleration	$a_c = \frac{v^2}{r}$
Centripetal Force	$F_c = m\frac{v^2}{r}$
Work	$W = Fd \cos \theta$
Kinetic Energy	$K = \frac{1}{2}mv^2$
Work-Energy Theorem	$W_{net} = \Delta K$
Power	$P = \frac{W}{t}$
Gravitational Potential Energy	$U_g = mgy$
Elastic Potential Energy	$U_s = \frac{1}{2}kx^2$
Work by Conservative Force	$W_c = -\Delta U$
Conservation of Mechanical Energy	$K_i + U_i = K_f + U_f$
Mechanical Energy with Nonconservative Work	$K_i + U_i + W_{nc} = K_f + U_f$