 # Torque Speed Review  ###### Jason Kuma

Writer | Coach | USC - Physics B.S & Business B.A. | Fremont, CA

You’ve covered linear forces, such as a push, pull, tension, weight, friction, etc. These forces cause a linear acceleration. Torque, on the other hand, is the rotational force that causes a rotational acceleration. By the end of this article you’ll learn how to apply a simple framework to solve any torque problem.

### Linear vs Rotational Equations

In Circular motion speed review, we covered rotational kinematics. Here we saw that all linear variables has a “rotational counterpart.” The chart below should help you give an idea on how to turn any linear concept into the rotational counterpart.

### Newton’s Second Law

The equation for torque is similar to Newton’s Second Law, Σ F = ma.

Torque = Σ τ = I𝜶, this tells us the SUM of all torques acting on a mass will cause rotation.

Just as a net linear force causes a linear acceleration, a net rotational force (Torque) causes a rotational acceleration (𝜶).

*Note: Angular acceleration and rotational acceleration are used interchangeably. They are both written in rad/s2 and mean the same thing.

Another 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) tells us to use ONLY the component of force applied perpendicularly to the lever arm.

In some cases, you will have to combine both equations to solve the problem:

I𝜶 = Fr(sinϴ)

#### Rotational Inertia

Note that rotational inertia is also referred to as a moment of inertia. It’s very similar to how we use mass, BUT it also involves the use of the radius of the object.

You do NOT need to memorize the equations below for the AP exam. And they will not be given on the formula sheet.

You DO need to know the rotational inertia for a point mass:

I point mass = mr2

*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.

Here some other rotational inertia formulas for other common objects.

#### Direction of Torque

Generally we stick to counter clockwise is positive and clockwise is negative.

An easy way to figure out direction. Take your pencil and push your finger in the direction of the force applied in the problem. See which way your pencil spins.

#### Parallel Axis Theorem

This is for AP Physics C students or higher.

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 = Icm + md2

• Icm 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. 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!

#### Static Equilibrium Questions (τnet = 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,

#### Dynamic Torque Questions (τnet ≠ 0 → 𝜶 ≠ 0))

• Finding the torque a pulley exerts on a system
• Finding the acceleration of a pulley system. Or finding the either tension in a pulley system
• Frictional torque acting on a system slows it down
• Finding angular acceleration of rotating platforms (like a merry-go-round) given torque

#### Rolling

• Objects, like spheres, will roll, not slide down ramps.
• Static friction causes a torque and rotation around the objects center.
• You will also need to apply conservation of energy and utilize rotational kinetic energy

### Solving 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 Speed Review.

Quick Recap So far

• 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.

#### Problem Solving Framework for Torque Problems

Follow this outline to solve any/all torque problem:

1.  Draw an FBD/diagram of the situation. You will want to also show the lever arm in the diagram.
2.  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. You measure the lever arm (r) from the pivot point.
3. Find the net Torque (τnet)of the object. Remember that τnet is simply the SUM of all torques around the pivot point. Note that static equilibrium = balanced torque → τnet = 0.
• You can use the formula T = Fr(sinϴ) to find each toruqe then add/subtract them based on the direction.
4. Lastly, set the net toque you found (τnet) equal to I𝜶 and solve your equation for the unknown value. Note in some cases the system is in static equilibrium, which means that there’s no angular acceleration. Thus I𝜶 = 0, because 𝜶 = 0. You still want to make an equation for τnet in any case.

### Rotational 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 as outlined in the table 1 above).

This will be covered in a separate post.

Since you are at the end of the AP Physics 1 curriculum, these questions were made with the AP exam in mind. There are NO numbers involved. Just variables and pure derivation will be used. All these questions are ranked > 8/10 in difficulty.

### (7) Five More Problems

##### (1) A 5-meter long ladder is leaning against a wall, with the bottom of the ladder 3 meters from the wall. The ladder is uniform and has a mass of 20 kg. A person of mass 80 kg is standing on the ladder at a distance of 4 meters from the bottom of the ladder. The ladder makes an angle of 60 degrees with the ground. What is the force exerted by the wall on the ladder?

⇨ concepts involved: Torque, linear forces, static torque
⇨ difficulty: medium

##### (2) A meter stick of mass 0.2 kg is pivoted at one end and supported horizontally. A force of 3 N is applied to the free end, perpendicular to the length of the meter stick. What is the net torque about the pivot point? (a) 0 (b) 1.50 (c) 2.48 (d) 3.00

⇨ concepts involved: net torque
⇨ difficulty: easy

##### (3) A meter stick of mass 200 grams is balanced at the 40-cm mark when a 100-gram mass is suspended from the 10-cm mark. What is the distance in centimeters from the pivot point to the center of mass of the meter stick?

⇨ concepts involved: static equilibrium, toque
⇨ difficulty: medium

##### (4) A horizontal uniform meter stick of mass 0.2 kg is supported at its midpoint by a pivot point. A mass of 0.1 kg is attached to the left end of the meter stick, and another mass of 0.15 kg is attached to the right end of the meter stick. The meter stick is free to rotate in the horizontal plane around the pivot point. What is the tension in the string supporting the left end of the meter stick?

⇨ concepts involved: non-static torque, tension
⇨ difficulty: hard

##### (5) A horizontal uniform rod of length L and mass M is pivoted at one end and is initially at rest. A small ball of mass M (same masses) is attached to the other end of the rod. The system is released from rest. What is the angular acceleration of the rod just immediately after the system is released? (a) g/L (b) 3g/4L (c) (m+1)g/L (d) (3mg)/(2L)

⇨ concepts involved: moment of inertia, torque
⇨ difficulty: medium

### Extra Help!

Need more help understanding these difficult topics? Do want to excel in Physics quickly? Book a 1-to-1 lesson with me and see the results for yourself. Over 400 AP, IB, and Honors students have joined so far. You can also join other Elite Physics programs here.

If you are a beginner, consider joining our Physics in 3 Weeks Program to get ahead and stay ahead of your classmates. See upcoming dates here. ###### Jason Kuma

Writer | Coach | USC - Physics B.S & Business B.A. | Fremont, CA

## Units in AP Physics 1

Unit 1 – Linear Kinematics

Unit 2 – Linear Forces

Unit 3 – Circular Motion

Unit 4 – Energy

Unit 5 – Momentum

Unit 6 – Torque

Unit 7 – Oscillations

# What makes the AP Physics exam so difficult?

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