Practice Quiz Study Guide: Newtonian Mechanics and Rotational Motion

Newtonian Mechanics

Fundamental Concepts

Newtonian mechanics forms the basis of classical physics, describing the motion of objects under the influence of forces. The following key principles and formulas are essential for solving problems in this domain.

• Newton's Laws of Motion:

  • First Law (Inertia): An object remains at rest or in uniform motion unless acted upon by a net external force.

  • Second Law: The net force on an object is equal to the mass times its acceleration. $\sum \vec{F} = m\vec{a}$

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

• Free-Body Diagrams (FBD): Visual representations of all forces acting on an object, crucial for problem-solving.

• Friction:

  • Static friction: $f_s \leq \mu_s N$

  • Kinetic friction: $f_k = \mu_k N$

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

• Inclined Planes: Forces must be resolved into components parallel and perpendicular to the surface.

Example Applications

• Elevator Tension: The tension in a cable supporting a mass in an accelerating elevator is $T = m(g + a)$ if accelerating upward, $T = m(g - a)$ if downward.

• Blocks on Frictionless Surface: When multiple blocks are pushed, the force transmitted between them depends on their masses and the applied force.

• Pushing Blocks with Friction: The acceleration is determined by the net force minus frictional force: $a = \frac{F_{\text{applied}} - f_k}{m}$

Rotational Motion

Key Concepts

Rotational motion describes the movement of objects around a fixed axis. The following are essential for understanding and solving rotational dynamics problems.

• Angular Displacement, Velocity, and Acceleration:

  • Angular displacement: $\theta$ (radians)

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

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

• Moment of Inertia: Quantifies an object's resistance to rotational acceleration about an axis.

  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: $K_{\text{rot}} = \frac{1}{2}I\omega^2$

• Torque: $\tau = rF\sin\theta$

• Newton's Second Law for Rotation: $\sum \tau = I\alpha$

• Angular Momentum: $L = I\omega$

Example Applications

• Rotating Rods and Strings: Analyzing tension and speed in rotating systems using FBDs and rotational dynamics.

• Vertical Loops (Amusement Park Rides):

  • Minimum normal force at the top: $N_{\text{min}} = 0$ (just enough to keep contact)

  • Maximum normal force at the bottom: $N_{\text{max}} = mg + \frac{mv^2}{R}$

Oscillatory Motion

Simple Harmonic Motion (SHM)

SHM describes systems where the restoring force is proportional to displacement, such as springs and pendulums.

• Spring-Mass System:

  • Equation of motion: $x(t) = A\cos(\omega t + \phi)$

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

  • Period: $T = 2\pi\sqrt{\frac{m}{k}}$

  • Energy: $E_{\text{total}} = \frac{1}{2}kA^2$

• Pendulum:

  • Period: $T = 2\pi\sqrt{\frac{l}{g}}$

Problem-Solving Strategies

General Approach

• Draw a clear free-body diagram (FBD) for each object.

• Identify all forces, including gravity, normal, tension, friction, and applied forces.

• Apply Newton's laws to set up equations for unknowns.

• For rotational problems, identify the axis of rotation and calculate moments of inertia.

• For systems with pulleys or multiple objects, consider constraints and relationships between accelerations and tensions.

• Check units and physical plausibility of your answers.

Numerical Constants

• Acceleration due to gravity: $g = 9.80\,\text{m/s}^2$

• Gravitational constant: $G = 6.67 \times 10^{-11}\,\text{N}\cdot\text{m}^2/\text{kg}^2$

Sample Multiple-Choice and Free-Response Questions

Multiple-Choice Examples

• Elevator Tension: Calculate the tension in a cable supporting a mass in an accelerating elevator.

• Force Transmission: Determine the force between blocks on a frictionless surface when a force is applied.

• Friction and Acceleration: Analyze the motion of a block under applied force and friction.

• Inclined Plane Forces: Resolve forces on a block on an inclined plane, including normal and spring scale readings.

• Pulley Systems: Compare accelerations in different pulley setups.

• Vertical Loop Forces: Find minimum and maximum normal forces in a rotating amusement park ride.

• Static Friction in Pulley Systems: Calculate the minimum mass required to prevent motion in a system with friction.

Free-Response Examples

• Rotating Rod and Strings:

  • Draw a free-body diagram for a mass attached to a rotating rod.

  • Given tension relationships, solve for the speed of the mass.

• Block on Inclined Plane:

  • Draw a free-body diagram for a block on a ramp.

  • Solve for the magnitude of force required to keep the block in equilibrium, expressing the answer in terms of $M$, $\theta$, and fundamental constants.

Summary Table: Moments of Inertia

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	$MR^2$

Additional info:

• Some formulas and constants are provided for reference during the quiz, including $g$ and $G$.

• Students are expected to show all work, including FBDs and step-by-step solutions for free-response questions.

• Quiz covers topics typical of a first-year college physics course: Newtonian mechanics, friction, rotational motion, and oscillatory motion.