Exam 2 Study Guide: Forces, Gravitation, Circular Motion, Work, and Energy

Newton's Second Law

One-Dimensional Problems

Newton's Second Law describes how the net force acting on an object determines its acceleration. In one dimension, this law is applied to objects moving along a straight line.

• Definition: Newton's Second Law states that the net force (Fnet) on an object is equal to the product of its mass (m) and its acceleration (a).

• Equation: $F_{\text{net}} = m a$

• Key Terms:

  • Acceleration: The rate of change of velocity with respect to time. $a = \frac{\Delta v}{\Delta t}$

  • Velocity: The speed of an object in a particular direction.

  • Speed: The magnitude of velocity; a scalar quantity.

• Example: A 2 kg object experiencing a net force of 10 N will have an acceleration of $a = \frac{F_{\text{net}}}{m} = \frac{10}{2} = 5 \text{ m/s}^2$.

Two-Dimensional Problems

When forces act in more than one direction, Newton's Second Law is applied to each component separately.

• Equation: $\vec{F}_{\text{net}} = m \vec{a}$

• Vector Components: Forces and accelerations are broken into x and y components.

• Example: An object subjected to forces in both x and y directions: $F_{\text{net},x} = m a_x$, $F_{\text{net},y} = m a_y$.

Universal Gravitation

Gravitational Field Strength and Acceleration Due to Gravity

The gravitational force between two masses is described by Newton's Law of Universal Gravitation. The constant g represents the acceleration due to gravity at the surface of a massive object.

• Equation: $F = G \frac{m_1 m_2}{r^2}$

• Gravitational Field Strength: $g = \frac{G M}{R^2}$ where M is the mass of the planet and R is its radius.

• Definition of Weight: The force of gravity acting on an object: $W = m g$

• Example: On Earth, $g \approx 9.8 \text{ m/s}^2$.

Density

Density is a measure of mass per unit volume.

• Equation: $\rho = \frac{m}{V}$

• Example: If a rock has mass 2 kg and volume 0.5 m3, its density is $\rho = \frac{2}{0.5} = 4 \text{ kg/m}^3$.

Forces with Names

Gravity, Normal Force, Tension, and Friction

Problems often involve multiple forces acting on one or more objects. Each force has a specific physical origin and mathematical description.

• Gravity: The force pulling objects toward the center of a massive body. $F_g = m g$

• Normal Force: The support force exerted by a surface perpendicular to the object.

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

• Friction: The force resisting motion between two surfaces. $F_{\text{friction}} = \mu F_{\text{normal}}$

• Multiple Objects: Apply Newton's Second Law to each object, considering all forces.

• Example: Two blocks connected by a rope, one on a table, one hanging: analyze forces and accelerations for both.

Applications of Newton's Second Law

Circular Motion: Car on a Banked Curve

Objects moving in a circle experience a centripetal force directed toward the center. For a car on a banked curve, the normal force and gravity combine to provide this force.

• Equation for Centripetal Force: $F_c = \frac{m v^2}{r}$

• Banked Curve: The angle of the bank allows part of the normal force to provide centripetal acceleration.

• Example: For a curve of radius r and bank angle θ, the ideal speed is $v = \sqrt{r g \tan \theta}$.

Resistive Force: Terminal Speed

When an object falls through air, it experiences a resistive force (drag) that increases with speed. Terminal speed is reached when the upward resistive force equals the downward gravitational force.

• Definition of Terminal Speed: The constant speed at which the net force on a falling object is zero.

• Equation: For quadratic drag, $F_{\text{drag}} = k v^2$ and terminal speed $v_t$ is found by setting $F_{\text{drag}} = F_g$:

• $m g = k v_t^2$

• $v_t = \sqrt{\frac{m g}{k}}$

• Example: A skydiver reaches terminal speed when air resistance balances weight.

Work

Definition and Calculation

Work is the transfer of energy by a force acting over a distance. The direction of force and displacement is important.

• Equation: $W = \vec{F} \cdot \vec{d} = F d \cos \theta$

• Dot Product: The dot product accounts for the angle between force and displacement vectors.

• Sign of Work: Positive if force and displacement are in the same direction; negative if opposite.

• Example: Lifting a box vertically: $W = m g h$

Conservation of Energy

Energy Changes and Work by Non-Conservative Forces

The principle of conservation of energy states that the total energy in a system remains constant unless acted upon by external forces. Work by non-conservative forces (like friction) changes the mechanical energy.

• Equation: $\Delta E = W_{\text{nc}}$

• Mechanical Energy: $E = K + U$ where K is kinetic energy and U is potential energy.

• Spring Example: An object accelerated by a horizontal spring:

  • Potential energy stored: $U = \frac{1}{2} k x^2$

  • Work done by spring: $W = \frac{1}{2} k x^2$

  • Change in kinetic energy: $\Delta K = W$

Summary Table: Key Forces and Their Equations

Force Name	Equation	Direction
Gravity	$F_g = m g$	Downward (toward center of mass)
Normal Force	Varies	Perpendicular to surface
Tension	Varies	Along string/rope
Friction	$F_{\text{friction}} = \mu F_{\text{normal}}$	Opposes motion
Drag (Air Resistance)	$F_{\text{drag}} = k v^2$	Opposes motion

Additional info: Academic context was added to clarify definitions, equations, and examples for each topic. The summary table was inferred to help organize the main forces discussed.