Torque PP Notes

Angle and Circular Motion

Angle ($\theta$): Related to radius (r) and arc length (s).
Radius (r): Distance from the center of a circle.
Arc Length (s): Distance along the outside of the circle.
Formula: \theta = \frac{s}{r} [m/m] which results in a unitless value (radians).
Revolution: 1 revolution = 2\pi radians = 360°.

Angular Velocity ($\omega$): Changing angle over time.
Formula: \omega = \frac{\Delta\theta}{t} [rad/s]
Example: Converting 45 rpm (revolutions per minute) to rad/s:
45 \frac{revolutions}{minute} \times \frac{2\pi \ radians}{1 \ revolution} \times \frac{1 \ minute}{60 \ s} = \omega
\omega = 45 \times \frac{2\pi}{60} = 4.7 \ rad/s

Angular Acceleration ($\alpha$): Changing angular velocity over time.
Formula: \alpha = \frac{\Delta\omega}{t} [rad/s^2]
\Delta\omega = \alpha t
\omegaf = \omegai + \Delta\omega
\omegaf = \omegai + \alpha t
Example: A wheel starts at rest and accelerates at 2.73 rad/s^2 for 4.87 s. What is the final angular velocity?
- \omega_i = 0 (starts at rest)
- \omega_f = 0 + 2.73 \times 4.87
- \omega_f = 13.3 \ rad/s

Rotational Inertia (I): Resistance to change in rotation; depends on mass and how it is distributed.
Formula for a point mass: I = mr^2 [kg m^2]

Torque ($\tau$): Causes rotation.
Formula: \tau = rF [Nm]
Relationship between angular acceleration, torque, and rotational inertia: \alpha = \frac{\tau}{I}
Linear acceleration: a = \frac{F}{m}
When \tau = 0, there is no angular acceleration.

Angular Momentum (L): Product of rotational inertia and angular velocity.
Formula: L = I\omega
Conservation of Angular Momentum: I1\omega1 = I2\omega2
- Ice Skater Example: An ice skater spins faster when they pull their arms in.
  - Extending arms out increases rotational inertia (mass further from the center).
  - Pulling arms in reduces rotational inertia.
  - Angular velocity increases because angular momentum remains constant.
Linear Momentum (p): p = mv

Everything on a carousel rotates at the same rate, meaning the same angular velocity (\omega).
Linear Velocity (v): Depends on the radius (r) and angular velocity.
Formula: v = r\omega

Angular momentum is conserved.
L = mrv
A comet moves faster when closer to the sun and slower when farther away, due to conservation of angular momentum.

When a car makes a turn, even if the speed (magnitude) stays the same, acceleration is required because the direction changes.
Centripetal Acceleration (a_c): Acceleration towards the center of the circle.
Formula: a_c = \frac{v^2}{r}
You feel an outward force (centrifugal force) due to your inertia trying to maintain a straight line.

Center of Gravity: The point where the weight of an object acts.
Equilibrium: Occurs when the center of gravity is over the point of support.
An object on its edge is at equilibrium but is unstable. Any movement causes torque, which accelerates it away from equilibrium.
Tipping Point: The point at which an object begins to fall over.

Center of Mass: The point around which an object rotates.
A thrown hammer rotates about its center of mass, while the center of mass follows a parabolic path.
The center of mass and center of gravity are slightly offset for the Moon.
Tidal Locking: The Moon is tidally locked with the Earth, meaning it rotates on its axis at the same rate it revolves around the Earth.

Circular Orbit: Achieved when an object falls towards Earth at the same rate it is curving.
Elliptical Orbit: Results from more energy than a circular orbit.
Escape Velocity: The minimum velocity required to break free of Earth’s gravity.
Black Hole: A large central mass that creates an escape velocity greater than the speed of light.