You’ve covered linear forces, such as a push, pull, tension, weight, friction, etc. Linear forces cause linear acceleration. Now we will cover **torque** or the force that accelerates an object *rotationally*.

### (1) Linear vs Rotational

Just as a linear force causes a linear acceleration, a rotational force (Torque) causes a rotational acceleration (alpha, symbol: 𝜶).

Remember that Torque applies the principles of Circular motion, just like forces apply the principles of linear motion (aka: kinematics). If you want to quickly review angular acceleration, check out the post “Circular Motion in 10 minutes.”

*Note: Angular acceleration and rotational acceleration are used interchangeably. They are both written in rad/s^{2} and mean the same thing.

### (2) Similarity in equations

The equation for torque is similar to F = ma.

Torque = **τ = ***I***𝜶**, where *I* (the letter “i”) is the rotational inertia, which is analogous to mass.

The other equation for Torque is **T = Fr(sinϴ)**

- where “F” is the applied force.
- “r” is the distance between the pivot point and the place where the force is applied. We call this the “lever arm”
- Note the
**(sinϴ)**just means that**only**the component of force applied perpendicularly will cause a torque. Most often the angle between r and F is 90° → sin(90°) = 1

In some cases, you will have to combine both equations to solve the problem: *I*𝜶 = Fr(sinϴ)

#### (2.1) Rotational Inertia

For objects that undergo angular acceleration, Rotational inertia ( *I* ) is the equivalent of mass. Note that Rotational inertia is also referred to as a *moment of inertia.*

The value for I is determined by (1) the shape of the object and (2) the point of rotation. Listed below are the formulas to find the rotational inertia for each object. You don’t need to memorize these. AP Physics C students might be asked to derive the formula for the rotational inertia of a disk/rod. This involves calculus, so be sure to know how to do this!

The most basic formula is * I = mr^{2}*. This is the rotational inertia for a point particle.

### (2.2) Parallel Axis Theorem

This is for advanced students/ AP Physics C students. The parallel axis theorem shows how to find the rotational inertia of a mass when the axis of rotation is **not **the center of mass. For example, a disk rotating 3/4 away from the center.

The parallel axis theorem is given as *I* = *I*_{cm} + md^{2}

*I*_{cm}is the rotational inertia of the center of mass.- d is the distance away from the center of mass
- m is the mass of the object

A common question type is when an object with velocity v sticks to another object (like a disk) and causes an angular acceleration. Another question is a roll of paper that unravels along the floor (thus does not rotates through the center).

### (3) How to Solve Torque Problems

Torque problems can be solved just like regular force problems. If you need a quick refresher on how to solve linear force problems skim through “Forces in 10 minutes.”

Quick notes:

- Instead of up and down, we use clockwise and counter-clockwise as directions.
- Generally, we say that clockwise is the negative direction
- Since toque is a force we can use it in combination with linear forces. This basically means we can have a system of 3 equations: 1 for torque (rotational direction), 1 for horizontal forces (linear-x-direction), and 1 for vertical direction (linear-y-direction)
- Most often friction between the axis of rotation and mass (such as a disk) is negligible. But when friction is
**not**negligible it produces a counter-torque, which causes an angular*deceleration*.

#### (3.1) General steps

Follow this outline to solve **any/all** torque problem:

- Draw an FBD/diagram of the situation.
- Identify the pivot point. This is simply the point around which the object rotates. For example, a rod of uniform mass tossed up will rotate about its midpoint.
- Find the net Torque (
**τ**)of the object. Remember that_{net}**τ**is simply the SUM of all torques around the pivot point. Note that static equilibrium = balanced torque → τ_{net}_{net}= 0. - Lastly, set
**τ**equal to_{net}and solve your equation for the unknown value. Note that in some cases the system is in static equilibrium, which means that there’s no angular acceleration. Thus*I*𝜶= 0, because*I*𝜶**𝜶**= 0.

### (4) Energy and Momentum

Energy and Momentum are others ways to solve toque problems!

As with linear energy and momentum, there is also rotational energy and momentum. The formulas are identical (just different variables).

This will be covered in different posts. For now, just focus on strictly using *forces *to solve problems as outlined above.

### (5) Types of torque problems

Here I’ll list the most common types of force problems you will see. It will be worthwhile studying these types of questions and understanding how to solve them. For more practice, there are 60 questions below to help you master torque!

#### (5.1) Static equilibrium questions (**τ**_{net} = 0 → 𝜶 = 0)

_{net}= 0 → 𝜶 = 0

- fulcrum and lever problems
- Tension of a slanted cable required to hold a horizontal platform
- Ladder problems, involving max weight on a ladder before slipping, required friction, etc,

#### (5.2) Dynamic torque questions (**τ**_{net} ≠ 0 → 𝜶 ≠ 0))

_{net}≠ 0 → 𝜶 ≠ 0

- Finding the torque a pulley exerts on a system
- Frictional torque acting on a system
- Finding angular acceleration of rotating platforms (like a merry-go-round) given torque.

### (6) 60 Questions to Master Torque (COMING SOON)

### (7) Extra Help!

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