Exam 2 Study Guide: Newton’s Laws, Gravitation, Circular Motion, Work, and Energy

Newton’s Second Law of Motion

One-Dimensional Problems

Newton’s Second Law relates the net force acting on an object to its acceleration. In one dimension, this law simplifies the analysis of motion along a straight line.

• Definition: The net force on an object is equal to the product of its mass and acceleration.

• Equation: $F_{\text{net}} = m a$

• Key Terms:

  • Acceleration (a): The rate of change of velocity with respect to time.

  • Velocity (v): The rate of change of displacement with respect to time.

  • Speed: The magnitude of velocity; always positive.

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

Two-Dimensional Problems

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

• Equation (vector form): $\vec{F}_{\text{net}} = m \vec{a}$

• Component Form: $F_{\text{net},x} = m a_x$, $F_{\text{net},y} = m a_y$

• Example: An object pulled at an angle requires resolving forces into x and y components before applying Newton’s Second Law.

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 acceleration due to gravity at the surface of a planet or other massive body is derived from this law.

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

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

• Definition of Weight: The gravitational force exerted on an object of mass m: $W = m g$

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

Density

• Definition: Density is mass per unit volume: $\rho = \frac{m}{V}$

Forces with Names

Applications Involving Multiple Forces

Problems often involve several named forces acting on one or more objects. Common forces include gravity, normal force, tension, and friction.

• Gravity (Weight): $W = m g$

• Normal Force: The perpendicular contact force exerted by a surface.

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

• Friction: The force resisting relative motion between surfaces. $f = \mu N$

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

• Example: Two blocks connected by a rope on a surface with friction require analyzing forces on each block and the tension in the rope.

Applications of Newton’s Second Law: Circular Motion and Resistive Forces

Circular Motion: Car on a Banked Curve

When a car moves on a banked curve, the normal force and friction provide the necessary centripetal force for circular motion.

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

• Banked Curve (no friction): $\tan \theta = \frac{v^2}{r g}$

• Example: Determining the banking angle required for a car to safely round a curve at a given speed without relying on friction.

Resistive Force: Terminal Speed

As an object falls through a fluid (like air), it experiences a resistive force that increases with speed. Eventually, the resistive force balances the gravitational force, and the object reaches terminal speed.

• Conceptual Definition: Terminal speed is the constant speed at which the net force on a falling object is zero (gravity is balanced by air resistance).

• Equation (for resistive force proportional to speed): $mg = bv_t$ so $v_t = \frac{mg}{b}$

• Example: A skydiver eventually stops accelerating and falls at a constant speed (terminal velocity).

Work

Definition and Calculation

Work is done when a force causes a displacement. The amount of work depends on the magnitude of the force, the displacement, and the angle between them.

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

• Dot Product: The dot product ensures that only the component of force in the direction of displacement does work.

• Sign of Work: Work is positive if the force has a component in the direction of displacement, negative if opposite.

• Example: Lifting a box upward (work done by gravity is negative, by you is positive).

Conservation of Energy

Energy Conservation and Non-Conservative Forces

The total mechanical energy (kinetic + potential) of a system remains constant if only conservative forces do work. Non-conservative forces (like friction) change the total mechanical energy.

• Conservation of Energy Equation: $K_i + U_i + W_{nc} = K_f + U_f$

• Kinetic Energy: $K = \frac{1}{2} m v^2$

• Potential Energy (Spring): $U_s = \frac{1}{2} k x^2$

• Work by Non-Conservative Forces: $W_{nc}$ is the work done by forces like friction or applied forces.

• Example: An object accelerated by a horizontal spring: energy stored in the spring is converted to kinetic energy and possibly work done against friction.

Force	Equation	Conservative?	Example
Gravity	$F_g = m g$	Yes	Object falling
Spring	$F_s = -k x$	Yes	Compressed/extended spring
Friction	$f = \mu N$	No	Sliding block
Tension	Varies	No	Hanging mass

Additional info: This guide expands on the study guide topics by providing definitions, equations, and examples for each concept, ensuring a comprehensive review for exam preparation.