Understanding Torque and Angular Acceleration

When we talk about how things spin, two important ideas come up: torque and angular acceleration. These ideas are connected by a rule from Newton, which is about how objects move when forces act on them.

The rule says:

$$\tau = I \alpha$$

Here, ( \tau ) is torque, ( I ) is the moment of inertia, and ( \alpha ) is angular acceleration. This means that the amount of torque on a spinning object affects how fast it speeds up as it spins.

1. What is Torque ($\tau$)?

• Torque is measured in Newton-meters (N·m).
• It depends on two things: the force you apply ($F$) and how far you are from the point it spins around (called the lever arm, $r$).

To find torque, we use this formula:

$$\tau = r \times F \sin(\theta)$$

In simpler terms, if you apply a force at a certain distance and angle, you can figure out the torque.

Example: If you push with a force of 10 N at a distance of 0.5 m straight out from the pivot point, you create a torque of 5 N·m.

2. What is Moment of Inertia ($I$)?

• Moment of inertia measures how mass is spread out around the axis where something spins.
• Here are some simple formulas for different shapes:
  • For a solid cylinder: ( I = \frac{1}{2} m r^2 )
  • For a solid sphere: ( I = \frac{2}{5} m r^2 )

3. What is Angular Acceleration ($\alpha$)?

• Angular acceleration is measured in radians per second squared (rad/s²).
• It happens because of the torque you apply, depending on the moment of inertia of the object.

In Summary

When you apply more torque to an object, it will spin faster, as long as its moment of inertia stays the same. This shows how torque and angular acceleration work together when things are in motion.