Article Content

Torque Speed Review

Picture of Jason Kuma
Jason Kuma

Writer | Coach | Builder | Fremont, CA

Get Expert Physics Coaching
Article Content
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.

Linear VariablesRotational Variables
∆x (displacement in meters )∆θ (angular displacement in radians)
∆v (velocity in m/s)∆ω (angular velocity in rad/s)
a (acceleration in m/s2)𝛂 (angular acceleration in rad/s2)
m (mass aka inertia)I (rotational inertia in kg·m2)
FormulasReplace linear variable with rotational counterparts
F = ma τ = I𝜶
KE = 1/2mv2KErotational = 1/2Iω2
p = mv → Linear momentumL = Iω → Angular momentum
∆p = Ft → Impulse∆L = τ(t) → Angular impulse
Table 1 – Linear vs Angular (Rotational) Variables.

Newton’s Second Law

The equation for torque is similar to Newton’s Second Law ( \sigma F = ma ). 

\text{Net Torque} = \Sigma \tau = I\alpha

This formula 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 ( \alpha ).

*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

\tau = Fr \cdot sin\theta

  • 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 sin\theta 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 like so: I\alpha = Frsin\theta

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 also will NOT be given on the formula sheet.

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

I_{\text{point mass}} = mr^2

*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 = I_{CM} + md^2 .

  • 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

When: \tau_{\text{net}} = 0 \rightarrow \alpha = 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

When: \tau_{\text{net}} \neq 0 \rightarrow \alpha \neq 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


  • 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 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 ( \tau_{Net} )of the object. Remember this is simply the SUM of all torques around the pivot point. And when in static equilibrium \tau_{Net} = 0.
    • You can use the formula \tau = Fr\cdot sin\theta to find each torque then add/subtract them based on the direction.
  4. Lastly, set the net toque you found ( \tau_{Net} )) equal to I\alpha 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 I\alpha = 0, because \alpha = 0. You still want to make an equation for \tau_{Net} 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 speed review covers rotational energy and momentum more in depth.

10 Challenging Problems to Help You Master Torque

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.

Question 1
Difficulty - Advanced
Solve Type - Mathematical
A uniform solid cylinder of mass M and radius R is initially at rest on a frictionless horizontal surface. A massless string is attached to the cylinder and is wrapped around it. The string is then pulled with a constant force F , causing the cylinder to rotate about its center of mass. After the cylinder has rotated through an angle \theta , what is the kinetic energy of the cylinder in terms of F and \theta ?
View Full Question and Explanation
Question 2
Difficulty - Intermediate
Solve Type - Mathematical
A seesaw is balanced on a fulcrum, with a boy of mass M_1 sitting on one end and a girl of mass M_2 sitting on the other end. The seesaw is a uniform plank of length L and mass M. The fulcrum is located at the midpoint of the plank. Does M_1 = M_2 . Justify your working.
View Full Question and Explanation
Question 3
Difficulty - Advanced
Solve Type - Mathematical
A boy is sitting at a distance d_1 from the fulcrum, and girl is sitting at a distance d_2 from the fulcrum, with d_1 > d_2 . The seesaw is level, with the two ends at the same height. Derive an equation for the minimum mass of the seesaw that will keep it balanced with the two children on it.
View Full Question and Explanation
Question 4
Difficulty - Advanced
Solve Type - Mathematical
A pulley system consists of two blocks of mass 5 kg and 10 kg, connected by a rope of negligible mass that passes over a pulley of radius 0.1 meters and mass 2 kg. The pulley is free to rotate about its axis. The system is released from rest, and the block of mass 10 kg starts to move downwards. Assuming that the coefficient of kinetic friction between the pulley and the rope is 0.2, and neglecting air resistance, determine
View Full Question and Explanation
Question 5
Difficulty - Advanced
Solve Type - Mathematical
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?
View Full Question and Explanation
Question 6
Difficulty - Beginner
Solve Type - Mathematical
A meter stick of mass .2 kg is pivoted at one end and supported horizontally. A force of 3 N downwards is applied to the free end, perpendicular to the length of the meter stick. What is the net torque about the pivot point?
  1. 0
  2. 1.5

  3. 2

  4. 3

  5. 4

View Full Question and Explanation
Question 7
Difficulty - Intermediate
Solve Type - Mathematical
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 from the pivot point to the center of mass of the meter stick? Give your answer in centimeters.
View Full Question and Explanation
Question 8
Difficulty - Advanced
Solve Type - Mathematical
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?
View Full Question and Explanation
Question 9
Difficulty - Intermediate
Solve Type - Mathematical
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?
  1. g/L

  2. 3g/4L

  3. (m+1)g/L

  4. (3mg)/(2L)

  5. None of these.

View Full Question and Explanation
Question 10
Difficulty - Advanced
Solve Type - Mathematical
A uniform solid sphere of mass M and radius R is placed on a frictionless horizontal surface. A massless string is wrapped around the sphere and is pulled with a force F. The string makes an angle of θ with the horizontal. What is the minimum value of the coefficient of static friction between the sphere and the surface required for the sphere to start rolling without slipping?
View Full Question and Explanation

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.

Picture of Jason Kuma
Jason Kuma

Writer | Coach | Builder | 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 

Unit 8 – Fluids

Reading Key


Nerd-Notes is free

and now you can help keep Nerd-Notes alive, remove all ads, get unlimited UBQ credits, access Phy Chat, and unlock 20 more features for the price of lunch.

Elite Tutoring for Physics

5 Weeks for a 5 on the AP Physics 1 Exam

Prepare for High School Physics

Nerd Notes

Discover the world's best Physics resources

Continue with

By continuing you (1) agree to our Terms of Sale and Terms of Use and (2) consent to sharing your IP and browser information used by this site’s security protocols as outlined in our Privacy Policy.

Made By
\Delta x = v_i t + \frac{1}{2} at^2F = ma
v = v_i + atF_g = \frac{G m_1m_2}{r^2}
a = \frac{\Delta v}{\Delta t}f = \mu N
R = \frac{v_i^2 \sin(2\theta)}{g} 
Circular MotionEnergy
F_c = \frac{mv^2}{r}KE = \frac{1}{2} mv^2
a_c = \frac{v^2}{r}PE = mgh
 KE_i + PE_i = KE_f + PE_f
MomentumTorque and Rotations
p = m v\tau = r \cdot F \cdot \sin(\theta)
J = \Delta pI = \sum mr^2
p_i = p_fL = I \cdot \omega
Simple Harmonic Motion
F = -k x
T = 2\pi \sqrt{\frac{l}{g}}
T = 2\pi \sqrt{\frac{m}{k}}
gAcceleration due to gravity, typically 9.8 , \text{m/s}^2 on Earth’s surface
GUniversal Gravitational Constant, 6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2
\mu_k and \mu_sCoefficients of kinetic (\mu_k) and static (\mu_s) friction, dimensionless. Static friction (\mu_s) is usually greater than kinetic friction (\mu_k) as it resists the start of motion.
kSpring constant, in \text{N/m}
M_E = 5.972 \times 10^{24} , \text{kg} Mass of the Earth
M_M = 7.348 \times 10^{22} , \text{kg} Mass of the Moon
M_M = 1.989 \times 10^{30} , \text{kg} Mass of the Sun
VariableSI Unit
s (Displacement)\text{meters (m)}
v (Velocity)\text{meters per second (m/s)}
a (Acceleration)\text{meters per second squared (m/s}^2\text{)}
t (Time)\text{seconds (s)}
m (Mass)\text{kilograms (kg)}
VariableDerived SI Unit
F (Force)\text{newtons (N)}
E, PE, KE (Energy, Potential Energy, Kinetic Energy)\text{joules (J)}
P (Power)\text{watts (W)}
p (Momentum)\text{kilogram meters per second (kgm/s)}
\omega (Angular Velocity)\text{radians per second (rad/s)}
\tau (Torque)\text{newton meters (Nm)}
I (Moment of Inertia)\text{kilogram meter squared (kgm}^2\text{)}
f (Frequency)\text{hertz (Hz)}

General Metric Conversion Chart

Example of using unit analysis: Convert 5 kilometers to millimeters. 

  1. Start with the given measurement: \text{5 km}

  2. Use the conversion factors for kilometers to meters and meters to millimeters: \text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}}

  3. Perform the multiplication: \text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}} = 5 \times 10^3 \times 10^3 \, \text{mm}

  4. Simplify to get the final answer: \boxed{5 \times 10^6 \, \text{mm}}



Power of Ten




















(Base unit)


Deca- or Deka-


















  1. Some answers may be slightly off by 1% depending on rounding, etc.
  2. Answers will use different values of gravity. Some answers use 9.81 m/s2, and other 10 m/s2 for calculations.
  3. Variables are sometimes written differently from class to class. For example, sometime initial velocity v_i is written as u ; sometimes \Delta x is written as s .
  4. Bookmark questions that you can’t solve so you can come back to them later. 
  5. Always get help if you can’t figure out a problem. The sooner you can get it cleared up the better chances of you not getting it wrong on a test!

Error Report

Sign in before submitting feedback.


Find 1000+ challenging AP Physics 1 problems to prep for your next exam.

It’s 100% free and Phy can help solve them all. 🤩

Try Prof Phy™

Phy can solve any question, grade your work, and simplify concepts. Just snap a picture.

Jason here! Feeling uneasy about your next physics test? We will help boost your grade in just two hours.

Thanks for reading Nerd-Notes.

Login or create a FREE account to continue reading.

By continuing, you agree to the updated Terms of Sale, Terms of Use, and Privacy Policy.

We use site cookies to improve your experience. By continuing to browse on this website, you accept the use of cookies as outlined in our privacy policy.