Review Problems: Ramps, Pulleys, and No-Mass Systems (Unit 4)

Ramps, Pulleys, and No-Mass Systems

Introduction

This study guide covers key concepts related to forces, friction, pulleys, and inclined planes, as commonly encountered in introductory college physics. The problems focus on applying Newton's Laws, analyzing equilibrium, and calculating acceleration and tension in various mechanical systems.

Forces and Newton's Laws of Motion

Friction and Motion on Flat Surfaces

Friction is a resistive force that opposes the relative motion of two surfaces in contact. The coefficient of kinetic friction ($\mu_k$) quantifies the frictional force between moving surfaces.

• Kinetic Friction Force: $f_k = \mu_k N$, where $N$ is the normal force.

• Application: When a tablecloth is pulled from under beakers, the beakers experience frictional force that determines their acceleration.

• Example: If beakers slide due to the tablecloth being pulled, their acceleration is $a = \mu_k g$.

Pulley Systems and Equilibrium

Analyzing Pulley Systems

Pulley systems are used to change the direction of forces and distribute loads. In equilibrium, the net force on each mass is zero.

• Equilibrium Condition: The sum of forces on each mass is zero.

• Example: If a blue object of mass $2.4\,\text{kg}$ is in equilibrium, the tension in the string equals the weight: $T = mg$.

• Height Calculation: If a green object is raised by pulling the string, the height moved is equal to the length of string pulled, considering the pulley arrangement.

Inclined Planes and Forces

Motion on Inclined Planes

Objects on inclined planes experience forces due to gravity, normal reaction, and applied forces. The acceleration depends on the net force along the incline.

• Force Components: The gravitational force along the incline is $mg \sin \theta$; the normal force is $mg \cos \theta$.

• Applied Force: An external force at an angle can be resolved into components parallel and perpendicular to the incline.

• Acceleration: $a = \frac{F_{\text{net}}}{m}$, where $F_{\text{net}}$ is the sum of all forces along the incline.

• Example: A $4.00\,\text{kg}$ block pulled up a $33.6^\circ$ incline by a $21.6\,\text{N}$ force at $18^\circ$ above the surface. Calculate net force and acceleration.

Combined Systems: Pulleys and Inclined Planes

Analyzing Multi-Object Systems

Systems with multiple objects connected by strings and pulleys require analyzing the forces on each object and the tension in the string.

• Free-Body Diagrams: Draw diagrams for each object, showing all forces (gravity, tension, friction).

• Direction of Motion: The system moves in the direction of the net force.

• Tension Calculation: Use Newton's Second Law for each object to solve for tension.

• Example: Object L ($5.00\,\text{kg}$) and Object R ($2.70\,\text{kg}$) connected over a frictionless pulley on an incline. Find acceleration and tension.

Sample Table: Forces on Inclined Plane System

Object	Mass (kg)	Forces Acting	Direction of Motion
L	5.00	Gravity, Tension, Friction	Up the incline
R	2.70	Gravity, Tension	Down the incline

Additional info: The table summarizes the forces and expected direction of motion for each object in a two-mass pulley system on an incline.

Key Equations

• Newton's Second Law: $F_{\text{net}} = ma$

• Kinetic Friction: $f_k = \mu_k N$

• Inclined Plane Force: $F_{\text{gravity, parallel}} = mg \sin \theta$

• Normal Force on Incline: $N = mg \cos \theta$

• Tension in String (Pulley): $T = m_1 a + m_1 g$ (for upward acceleration)

Summary Table: Problem Answers

Problem	Answer
1	$a = \mu_k g$
2	2.4 kg; 12 cm
3	$a = 0.34\,\text{m/s}^2$
4	$a = 2.1\,\text{m/s}^2$
5	Object L slips uphill, Object R slips downhill; $a = 1.35\,\text{m/s}^2$, $T = 46.9\,\text{N}$

Additional info: These answers correspond to the review problems and demonstrate the application of Newton's Laws, friction, and tension calculations in mechanical systems.