Physics Study Guide: Newton's Laws, Circular Motion, Work, and Energy (Chapters 5–8)

Newton's Laws of Motion

Newton's Third Law

Newton's Third Law states that for every action, there is an equal and opposite reaction. This principle is fundamental in analyzing interactions between objects.

• Definition: If object A exerts a force on object B, then object B exerts a force of equal magnitude and opposite direction on object A.

• Application: When two objects interact (e.g., a book resting on a table), the book pushes down on the table, and the table pushes up on the book with an equal force.

• Equation: $\vec{F}_{A \rightarrow B} = -\vec{F}_{B \rightarrow A}$

• Example: When you jump off a boat, you push the boat backward as you move forward.

Circular Motion

Path, Angle, and Tension in Circular Motion

Objects moving in a circle experience a net force directed toward the center of the circle, called the centripetal force. The tension in a string or the normal force can provide this centripetal force.

• Path: The trajectory is a circle with radius r.

• Angle: The angle often refers to the orientation of the force or the position of the object along the circle.

• Tension: For an object of mass m moving at speed v in a circle of radius r:

$F_{c} = \frac{mv^2}{r}$

• Example: A ball on a string swung in a horizontal circle; the tension in the string provides the centripetal force.

Mass and Weight

Difference Between Mass and Weight

Mass and weight are related but distinct physical quantities.

• Mass (m): A measure of the amount of matter in an object; SI unit is the kilogram (kg).

• Weight (W): The force of gravity acting on an object; depends on the local gravitational acceleration (g).

$W = mg$

• Example: An object with mass 2 kg has a weight of 19.6 N on Earth ($g = 9.8\ \mathrm{m/s^2}$).

Motion Under Gravity

Comparing Objects in Free Fall

Objects moving under the influence of gravity alone (free fall) experience the same acceleration regardless of mass (neglecting air resistance).

• Acceleration: $a = g = 9.8\ \mathrm{m/s^2}$ downward near Earth's surface.

• Key Point: All objects fall at the same rate in a vacuum.

• Example: A feather and a hammer dropped on the Moon fall at the same rate.

Equilibrium and Force Components

Forces on an Object in Equilibrium

An object is in equilibrium when the net force acting on it is zero. Forces can be resolved into components, typically along the x and y axes.

• Equilibrium Condition: $\sum F_x = 0$, $\sum F_y = 0$

• Resolving Forces: Use trigonometry to find components (e.g., $F_x = F \cos \theta$, $F_y = F \sin \theta$).

• Example: A block at rest on an inclined plane; the normal force and friction balance the component of gravity.

Newton's Second Law and Kinematics

Combining Newton's Second Law with Kinematic Equations

Newton's Second Law relates the net force on an object to its acceleration. Kinematic equations describe motion with constant acceleration.

• Newton's Second Law: $\vec{F}_{\text{net}} = m\vec{a}$

• Kinematic Equations:

$\begin{align*} v &= v_0 + at \\ x &= x_0 + v_0 t + \frac{1}{2} a t^2 \\ v^2 &= v_0^2 + 2a(x - x_0) \end{align*}$

• Example: Finding the acceleration of a block sliding down a frictionless incline and its speed after 2 seconds.

Work and Energy

Work Done by Various Forces

Work is done when a force causes displacement. Different forces (gravity, springs, friction) do work in different ways.

• Work by a Constant Force: $W = Fd \cos \theta$

• Work by Gravity: $W_g = mg(h_1 - h_2)$

• Work by a Spring: $W_s = \frac{1}{2}k(x_i^2 - x_f^2)$

• Work by Friction: $W_f = -f_k d$

• Example: Calculating the work done by friction as a box slides across a floor.

Kinetic Energy

Kinetic energy is the energy of motion, dependent on mass and velocity.

• Formula: $KE = \frac{1}{2}mv^2$

• Example: A 3 kg object moving at 4 m/s has $KE = 24$ J.

Conservation of Mechanical Energy

The total mechanical energy (kinetic + potential) of an object remains constant if only conservative forces (like gravity) do work.

• Principle: $KE_i + PE_i = KE_f + PE_f$

• Potential Energy (Gravity): $PE = mgh$

• Example: A ball dropped from a height; potential energy converts to kinetic energy as it falls.

Work-Kinetic Energy Principle

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

• Equation: $W_{\text{net}} = \Delta KE = KE_f - KE_i$

• Example: Calculating the final speed of a car after a net force acts over a distance.

Inclined Planes and Free-Body Diagrams

Drawing Free-Body Diagrams and Resolving Forces

Free-body diagrams help visualize all forces acting on an object. On an inclined plane, forces are resolved into components parallel and perpendicular to the surface.

• Steps:

  1. Draw the object and all forces acting on it (gravity, normal force, friction, applied force).

  2. Resolve gravity into components: parallel ($mg \sin \theta$) and perpendicular ($mg \cos \theta$) to the incline.

  3. Apply Newton's Second Law in the x (along the incline) and y (perpendicular) directions.

• Example: Finding the acceleration of a block sliding down a rough incline.