PHYS 151 Midterm Exam #4 Study Guide: Rotational Motion, Dynamics, and Fluid Mechanics

Rotational Motion and Dynamics

Angular Quantities and Unit Conversions

Rotational motion involves the movement of objects around a fixed axis. Key quantities include angular displacement, angular velocity, and angular acceleration, which are analogous to their linear counterparts.

• Angular Displacement (θ): The angle through which an object rotates, measured in radians.

• Angular Velocity (ω): The rate of change of angular displacement, measured in radians per second ($\omega = \frac{d\theta}{dt}$).

• Angular Acceleration (α): The rate of change of angular velocity ($\alpha = \frac{d\omega}{dt}$).

• Unit Conversion: 1 revolution = $2\pi$ radians; 1 radian = 57.3 degrees.

Example: Converting revolutions per minute (rpm) to radians per second.

• Given: 33.3 rpm

• Conversion: $33.3 \text{ rpm} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} = 3.49 \text{ rad/s}$

Moment of Inertia

The moment of inertia quantifies an object's resistance to changes in rotational motion. It depends on the mass distribution relative to the axis of rotation.

• Definition: $I = \sum m_i r_i^2$ for discrete masses, or $I = \int r^2 dm$ for continuous bodies.

• Common Shapes:

  • Solid cylinder: $I = \frac{1}{2}MR^2$

  • Thin rod (center): $I = \frac{1}{12}ML^2$

  • Thin rod (end): $I = \frac{1}{3}ML^2$

• Parallel Axis Theorem: $I = I_{cm} + Md^2$, where $d$ is the distance from the center of mass axis.

Example: Calculating the moment of inertia for a ceiling fan or a balance pole.

Rotational Kinetic Energy

Rotating objects possess kinetic energy due to their angular motion.

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

• Change in Rotational Kinetic Energy: $\Delta K_{rot} = \frac{1}{2}I(\omega_f^2 - \omega_i^2)$

Example: Calculating the change in kinetic energy when a fan speeds up.

Torque and Rotational Dynamics

Torque is the rotational equivalent of force and causes changes in angular velocity.

• Definition: $\tau = rF\sin\theta$

• Newton's Second Law for Rotation: $\tau_{net} = I\alpha$

• Power Delivered: $P = \tau \omega$

Example: Calculating the torque and power required to accelerate a rotating object.

Conservation of Angular Momentum

Angular momentum is conserved in the absence of external torques.

• Definition: $L = I\omega$

• Conservation Principle: $I_i\omega_i = I_f\omega_f$

Example: A figure skater pulling in arms to spin faster.

Applications: Ceiling Fans and Balance Poles

Ceiling Fan Dynamics

Ceiling fans are practical examples of rotational motion, involving calculations of angular velocity, frequency, and energy.

• Frequency: $f = \frac{\omega}{2\pi}$

• Rotational Kinetic Energy: See above.

• Power Delivered: $P = \tau \omega$

Example: Determining the energy and power required to accelerate a fan from rest.

Tightrope Walker and Balance Pole

Balance poles increase a walker's moment of inertia, helping maintain stability by reducing angular acceleration for a given torque.

• Moment of Inertia for a Rod: $I = \frac{1}{12}ML^2$ (center)

• Effect: Larger $I$ means slower rotation, aiding balance.

Example: Calculating the moment of inertia and angular acceleration for a balance pole.

Fluid Mechanics

Buoyancy and Archimedes' Principle

Buoyancy is the upward force exerted by a fluid on a submerged object, described by Archimedes' Principle.

• Archimedes' Principle: The buoyant force equals the weight of the fluid displaced.

• Formula: $F_b = \rho_{fluid} V_{displaced} g$

• Density: $\rho = \frac{m}{V}$

Example: Calculating the buoyant force on a cooler submerged in water.

Volume and Density Calculations

Determining the volume and density of objects is essential for analyzing buoyancy and equilibrium in fluids.

• Volume of a Rectangular Object: $V = l \times w \times h$

• Density: $\rho = \frac{m}{V}$

Example: Finding the volume of a cooler and its density after water enters.

Equilibrium and Floating Conditions

Objects float when their average density is less than the fluid's density. Equilibrium occurs when the buoyant force equals the object's weight.

• Condition for Floating: $\rho_{object} < \rho_{fluid}$

• Equilibrium: $F_b = mg$

Example: Calculating the final mass and density of a cooler after water enters.

Summary Table: Key Rotational and Fluid Quantities

Quantity	Symbol	Formula	Units
Angular Velocity	$\omega$	$\omega = \frac{d\theta}{dt}$	rad/s
Moment of Inertia	$I$	$I = \sum m_i r_i^2$	kg·m2
Rotational Kinetic Energy	$K_{rot}$	$K_{rot} = \frac{1}{2}I\omega^2$	J
Torque	$\tau$	$\tau = rF\sin\theta$	N·m
Buoyant Force	$F_b$	$F_b = \rho_{fluid} V_{displaced} g$	N
Density	$\rho$	$\rho = \frac{m}{V}$	kg/m3

Additional info:

• Some questions reference the Parallel Axis Theorem and rotational equilibrium, which are essential for analyzing composite systems.

• Applications include real-world objects such as ceiling fans, balance poles, and coolers, illustrating the practical relevance of rotational and fluid mechanics.