Physics I: Mechanics – Quiz Study Guide (Forces, Motion, Energy, Rotational Dynamics)

Quiz Instructions and Constants

General Guidelines

This quiz covers fundamental topics in introductory physics, focusing on mechanics. Students are required to answer multiple-choice and free-response questions, showing all work for full credit. Key constants are provided for reference.

• g = 9.80 m/s2 (acceleration due to gravity)

• G = 6.67 × 10-11 N·m2/kg2 (universal gravitational constant)

Kinematics and Dynamics

Equations of Motion

Kinematics describes the motion of objects using equations that relate displacement, velocity, acceleration, and time.

• Displacement: $\Delta x = v_{avg} \Delta t$

• Velocity: $v = v_0 + at$

• Displacement (with constant acceleration): $\Delta x = v_0 t + \frac{1}{2} a t^2$

• Final velocity squared: $v^2 = v_0^2 + 2a \Delta x$

These equations are used to solve problems involving linear motion, such as objects sliding down inclines or moving under the influence of gravity.

Newton's Laws and Free-Body Diagrams

Newton's laws of motion form the foundation of classical mechanics. Free-body diagrams (FBDs) are essential tools for visualizing forces acting on an object.

• Newton's First Law: An object remains at rest or in uniform motion unless acted upon by a net external force.

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

• Newton's Third Law: For every action, there is an equal and opposite reaction.

• Free-Body Diagram: A graphical representation showing all forces acting on an object, such as gravity, normal force, tension, friction, and applied forces.

Example: Analyzing a block on an inclined plane involves resolving forces into components parallel and perpendicular to the surface.

Forces on Inclined Planes

Components of Forces

When an object rests on an inclined plane, the gravitational force is resolved into two components:

• Parallel to the incline: $F_{\parallel} = mg \sin \theta$

• Perpendicular to the incline: $F_{\perp} = mg \cos \theta$

The normal force equals $F_{\perp}$ if no other vertical forces are present. Tension and friction may also act depending on the setup.

Example Problem

A mass $M$ is held at rest on a frictionless ramp inclined at angle $\theta$ by a string. The tension in the string balances the component of gravity parallel to the ramp.

• Tension: $T = mg \sin \theta$

• Normal force: $N = mg \cos \theta$

• Acceleration down the ramp (if string breaks): $a = g \sin \theta$

Friction

Kinetic and Static Friction

Friction opposes the relative motion between surfaces. The coefficient of friction quantifies the strength of this force.

• Kinetic friction: $f_k = \mu_k N$

• Static friction: $f_s \leq \mu_s N$

Example: Calculating the coefficient of kinetic friction for a car skidding to a stop involves energy conservation and the work done by friction.

Work and Energy

Work, Kinetic Energy, and Potential Energy

Work is the transfer of energy via force acting over a distance. The work-energy theorem relates work to changes in kinetic energy.

• Work: $W = \int \vec{F} \cdot d\vec{s}$

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

• Potential energy (gravity): $U_g = mgh$

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

• Work-energy theorem: $W_{net} = \Delta K$

Example: A spring attached to a mass on an incline stretches until the spring force balances the component of gravity parallel to the incline.

Rotational Motion and Dynamics

Angular Kinematics and Dynamics

Rotational motion involves angular displacement, velocity, and acceleration. The moment of inertia quantifies an object's resistance to rotational acceleration.

• Angular displacement: $\theta$ (radians)

• Angular velocity: $\omega = \frac{d\theta}{dt}$

• Angular acceleration: $\alpha = \frac{d\omega}{dt}$

• Rotational analog of Newton's second law: $\sum \tau = I \alpha$

Moment of Inertia

The moment of inertia depends on the mass distribution relative to the axis of rotation.

Object	Moment of Inertia
Rod (through CM)	$(1/12)ML^2$
Rod (through end)	$(1/3)ML^2$
Solid sphere	$(2/5)MR^2$
Hollow sphere	$(2/3)MR^2$
Solid disk, solid cylinder	$(1/2)MR^2$
Ring/hoop, hollow cylinder	$MR^2$

Rotational Kinetic Energy and Dynamics

• Rotational kinetic energy: $K_{rot} = \frac{1}{2} I \omega^2$

• Torque: $\tau = r F \sin \phi$

• Angular momentum: $L = I \omega$

Example: Calculating the normal force on a car moving over a hill or through a dip involves centripetal acceleration and Newton's second law.

Oscillations and Gravitation

Simple Harmonic Motion

Oscillatory motion occurs when a restoring force is proportional to displacement, such as a mass on a spring.

• Period of a mass-spring system: $T = 2\pi \sqrt{\frac{m}{k}}$

• Displacement as a function of time: $x(t) = A \cos(\omega t + \phi)$

• Angular frequency: $\omega = \sqrt{\frac{k}{m}}$

Universal Gravitation

• Gravitational force: $F_g = \frac{G m_1 m_2}{r^2}$

• Gravitational potential energy: $U = -\frac{G m_1 m_2}{r}$

Free-Body Diagram Practice

Drawing and Interpreting FBDs

Free-body diagrams are used to analyze forces in various scenarios, such as blocks on surfaces, objects on inclines, and systems with multiple interacting bodies.

• Identify all forces: gravity, normal, tension, friction, applied forces.

• Label each force with its type and direction.

• For combined systems, consider the net force and interactions between components.

Example: For a cart and block system, draw separate FBDs for each object and for the combined system, indicating all relevant forces.

Summary Table: Key Formulas

Topic	Formula
Kinematics	$v = v_0 + at$
Forces	$\sum F = ma$
Friction	$f_k = \mu_k N$
Work	$W = Fd \cos \theta$
Kinetic Energy	$K = \frac{1}{2} m v^2$
Potential Energy	$U_g = mgh$
Spring Force	$F_s = -kx$
Rotational Motion	$\sum \tau = I \alpha$
Moment of Inertia	See table above
Gravitational Force	$F_g = \frac{G m_1 m_2}{r^2}$

Additional info:

• Some questions require interpreting diagrams and applying principles to new situations, such as equilibrium on an incline, energy conservation, and rotational dynamics.

• Students should be familiar with both conceptual reasoning and quantitative problem-solving.