Rotational Motion Practice Flashcards

Introduction to Rotational Motion

• Rotational motion is defined as the motion of objects that rotate around an axis. This is in contrast to translational motion, which refers to motion along a straight line.

• Standard concepts of translational motion are insufficient to fully describe or understand complex systems such as bicycle wheels, bowling balls, figure skaters, propellers, or the motion of heavenly bodies like stars and planets.

• To analyze rotational motion, several key physical quantities must be established: rotational speed, tangential velocity, rotational acceleration, rotational inertia, torque, center of mass, stability, angular momentum, centripetal force, and centripetal acceleration.

Rotational Speed ($\omega$)

• Rotational speed measures how quickly or slowly an object rotates about an axis of rotation.

• The symbol for rotational speed is the lowercase Greek letter omega ($\omega$).

• Units for rotational speed vary based on the method of measurement:

  • Revolutions/Rotations: Measured as the number of complete turns per unit of time (e.g., revolutions per minute, or $\text{RPM}$).

  • Degrees: Measured as the angular distance in degrees covered per unit of time. One full rotation is equal to $360^\circ$.

  • Radians: Measured as the angular distance in radians covered per unit of time. One full rotation is equal to $2\pi\,\text{radians}$.

• Case Study: Geostationary Satellites

  • A geostationary satellite orbits the Earth at a distance and speed that allows it to remain stationary relative to an observer on the ground.

  • It completes one orbit every $24\,\text{hours}$.

  • Calculating rotational speed in $\text{RPM}$: $24\,\text{hours}$ is equal to $1440\,\text{minutes}$. The rotational speed is $\frac{1\,\text{revolution}}{1440\,\text{min}} = 6.94 \times 10^{-4}\,\text{rev/min}$.

  • Calculating rotational speed in degrees per second: One orbit ($360^\circ$) takes $86440\,\text{seconds}$. The rotational speed is $\frac{360^\circ}{86440\,\text{s}} = 0.004^\circ/\text{s}$.

  • Calculating rotational speed in radians per second: One orbit ($2\pi\,\text{rad}$) takes $86440\,\text{seconds}$. The rotational speed is $\frac{2\pi\,\text{rad}}{86440\,\text{s}} = 7.27 \times 10^{-5}\,\text{rad/s}$.

Tangential Velocity ($v_T$)

• Tangential velocity describes the linear speed of an object moving along a circular path. While rotational speed ($\omega$) remains constant for all points on a rigid rotating body, tangential velocity varies based on the distance from the center of rotation.

• The symbol for tangential velocity is $v$ (or $v_T$), and the standard unit is meters per second ($\text{m/s}$).

• The magnitude of tangential velocity is calculated using the circumference of the circular path ($2\pi r$) divided by the time required for one rotation ($T$):

  • $v_T = \frac{2\pi r}{T}$

• The relationship between distance and speed is direct: the farther an object is from the axis of rotation ($r$), the higher its tangential speed. Conversely, the closer it is to the center, the smaller the tangential speed.

• The direction of the tangential velocity is always tangent to the circle of rotation. An example provided is the direction an object would travel if released from a spinning slingshot.

• Demonstration: Bicycle Wheel

  • Pieces of tape were placed at different radii on a wheel rotating with a period of $3.07\,\text{s}$.

  • White Tape: Positioned $26.5\,\text{cm}$ ($0.265\,\text{m}$) from the center. It travels a linear distance (circumference) of $1.67\,\text{m}$ in one revolution. Its tangential speed is $0.54\,\text{m/s}$.

  • Purple Tape: Positioned $9\,\text{cm}$ ($0.09\,\text{m}$) from the center. It travels a linear distance of $0.57\,\text{m}$ in one revolution. Its tangential speed is $0.18\,\text{m/s}$.

  • Both tapes share the same rotational speed ($\text{RPM}$, degrees/s, or rad/s) despite different tangential speeds.

Rotational Acceleration

• Acceleration in rotational contexts occurs when an object speeds up, slows down, or changes the direction of its rotation.

• While not the primary focus of the lecture, it is noted that just as force causes changes in linear motion, torque causes changes in rotational motion (rotational acceleration).

Rotational Inertia

• Rotational inertia is a measure of an object's resistance to changes in its rotational state. It determines how difficult it is to start an object rotating or to change the speed of an object already in motion.

• It depends on two factors:

  1. The mass ($m$) of the object.

  2. The distribution of that mass relative to the axis of rotation.

• Mass Distribution Principle: The farther the mass is distributed from the axis of rotation, the higher the rotational inertia.

• Example: Hammer Experiment

  • Rotating a hammer by the head: The mass is concentrated near the hand (axis), resulting in low rotational inertia. Rotation is easy.

  • Rotating a hammer by the handle: The heavy head is now far from the axis, significantly increasing rotational inertia. This makes it much harder to rotate back and forth.

• Example: Hoop vs. Solid Disk

  • A hollow hoop and a solid disk have equal radii and nearly identical masses.

  • Because the hoop's mass is concentrated at its outer edge (far from the center), it has a higher rotational inertia than the solid disk, whose mass is distributed throughout.

  • In a race down a ramp, the disk reaches the bottom first because the hoop's higher rotational inertia makes it more resistant to starting and maintaining rotation.

Torque ($\tau$)

• Torque is a twist or turn that causes an object to change its rotational motion. It is the rotational equivalent of force.

• The symbol for torque is the lowercase Greek letter tau ($\tau$), and the units are Newton-meters ($\text{N}\cdot\text{m}$). Note that while this unit is dimensionally equivalent to Joules, Joules is reserved for work and energy, not torque.

• The magnitude of torque depends on three factors:

  1. Force Magnitude ($F$): Torque is directly proportional to the applied force.

  2. Lever Arm Distance ($r$): Torque is proportional to the distance from the rotational axis to the point where force is applied.

  3. Angle of Force: Only the perpendicular component of the force ($F_\perp$) contributes to torque. Parallel components do not affect rotation.

• Formula: $\tau = F_\perp \times r$

• Application: Doors and Wrenches

  • Opening a door is most efficient when pushing perpendicularly at the edge farthest from the hinges. This maximizes the lever arm ($r$) and ensures a large perpendicular force component.

  • Doorknobs are placed at the opposite end of the door from the hinges specifically to maximize the lever arm.

  • Socket wrench extensions (cheater bars) allow users to apply more torque to tight bolts by increasing the lever arm distance when their physical strength (force) is limited.

• Example Calculation 1 (Meter Stick):

  • A $10\,\text{g}$ ($0.01\,\text{kg}$) mass is placed at a distance of $15\,\text{cm}$ ($0.15\,\text{m}$) from a fulcrum.

  • Force ($F_g$): $0.01\,\text{kg} \times 10\,\text{m/s}^2 = 0.1\,\text{N}$.

  • Torque: $0.1\,\text{N} \times 0.15\,\text{m} = 0.015\,\text{N}\cdot\text{m}$.

• Example Calculation 2 (Non-Perpendicular Force):

  • A force vector of $5\,\text{N}$ acts at a distance of $20\,\text{cm}$ ($0.2\,\text{m}$).

  • The components are $4\,\text{N}$ (perpendicular/down) and $3\,\text{N}$ (parallel/right).

  • Only the $4\,\text{N}$ component is used: $\tau = 4\,\text{N} \times 0.2\,\text{m} = 0.8\,\text{N}\cdot\text{m}$.

Mechanical Equilibrium

• Mechanical equilibrium is reached when an object maintains a constant linear velocity and a constant rotational speed.

• Conditions for Equilibrium:

  1. The net force acting on the object must be zero ($\sum F = 0$).

  2. The net torque acting on the object must be zero ($\sum \tau = 0$).

• Directional Sign Convention for Torque:

  • Counterclockwise rotations are generated by positive torque.

  • Clockwise rotations are generated by negative torque.

Center of Mass (CM) and Center of Gravity (CG)

• Center of Mass (CM): The specific point in an object around which it rotates. It represents the average position of all the mass in the system. If supported at the CM, an object will balance perfectly.

  • In uniform objects, the CM is at the geometric center (e.g., center of a meter stick).

  • In non-uniform objects, the CM shifts toward the area of higher mass concentration (e.g., near the head of a hammer).

  • The CM can exist in a location where there is no physical material, such as the hollow center of a metal hoop.

• Center of Gravity (CG): The point that represents the average distribution of weight.

  • For most objects, CM and CG are identical because the force of gravity is uniform across the object.

  • Exceptions include extremely tall structures like the Burj Khalifa ($830\,\text{m}$ tall). Because gravity is stronger closer to Earth's surface, the CG of the Burj Khalifa is approximately a couple of millimeters below its CM.

Stability

• Stability refers to how easily an object will topple over when disturbed from equilibrium.

• Condition for Stability: An object is stable if its center of mass is supported directly above its base. If a disturbance moves the CM outside of the support base, gravity will create a torque that causes the object to topple.

• Example: Leaning Tower Demo

  • A tower remains upright as long as its CM is over the base. Rotating a top-mounted weight can shift the tower's CM outside the base area, causing it to fall.

• Example: Soda Can

  • A full or empty can has a high CM. By leaving a specific amount of liquid inside, the CG is lowered, allowing the can to balance on its bottom edge. This configuration is fragile because the resulting support base is very small.

• Application: Aviation

  • Pilots must calculate the distribution of passengers and cargo to ensure the aircraft's CG remains within specific limits for stable flight. Seat reassignments are sometimes required to maintain this balance.

Angular Momentum ($L$)

• Momentum is defined as inertia in motion; angular momentum is rotational inertia in rotational motion.

• Equation: $\text{Angular Momentum} = \text{Rotational Inertia} \times \text{Rotational Speed}$.

• Conservation of Angular Momentum: In the absence of an external net torque, the angular momentum of a system remains constant.

• Demonstration: Rotating Chair

  • A person spinning on a chair with dumbbells can change their rotational speed by altering their mass distribution.

  • Extending arms increases rotational inertia, causing speed to decrease to keep momentum conserved.

  • Pulling arms in reduces rotational inertia, causing speed to increase.

  • This effect is commonly seen in figure skaters during spins.

Centripetal Force ($F_C$)

• Any object moving in a circular path at a constant speed is accelerating because its direction is constantly changing. The force that causes this is centripetal force.

• The symbol is $F_C$ and the unit is Newtons ($\text{N}$).

• Formula: $F_C = \frac{m v^2}{r}$

  • $m$ is mass.

  • $v$ is tangential velocity.

  • $r$ is the radius of the path.

• Centripetal force is not a standalone force but must be provided by another mechanism (e.g., tension, gravity, friction, or normal force).

  • Car turning: Friction between tires and road provides $F_C$. Slippery roads reduce available friction, making turns harder.

  • Satellites: Gravity provides the centripetal force for orbit.

  • BB in coiled tubing: The normal force from the walls of the tube provides $F_C$.

• Example Calculation (BB):

  • Mass ($m$) = $0.33\,g = 0.00033\,kg$.

  • Radius ($r$) = $10.25\,cm = 0.1025\,m$.

  • Tangential speed ($v$) = $0.46\,m/s$.

  • Time for path = $1.4\,s$.

  • $F_C = \frac{0.00033 \times (0.46)^2}{0.1025} = 6.8 \times 10^{-4}\,\text{N}$.

Centripetal Acceleration ($a_C$)

• Derived from Newton's second law ($F = ma$), centripetal acceleration is the direction-changing acceleration pointing toward the center of a circular path.

• Formula: $a_C = \frac{v^2}{r}$

• Demonstration: Cup of Water on a String

  • When a cup of water is swung in a vertical circle, the water stays inside due to inertia. For the water to stay in the cup at the top of the loop, the centripetal acceleration must be greater than the acceleration due to gravity ($g = 9.8\,m/s^2$).

  • Radius ($r$) = $0.46\,m$.

  • Period ($T$) = $0.63\,s$.

  • Tangential speed ($v$) = $4.59\,m/s$.

  • $a_C = \frac{(4.59)^2}{0.46} = 45.75\,m/s^2$.

  • In terms of g-force: $\frac{45.75}{9.8} = 4.67\,g's$. This confirms the water will stay in the cup because the cup is accelerating at over four times the rate of gravity.