--- title: "Rotational Kinematics and Torque Speed Review" description: "AP Physics. Review Torque in 10 minutes or less. This is every thing you need to know + 60 AP style questions to MASTER torque." featured_image: "https://nerd-notes.com/wp-content/uploads/2022/07/tekton-HmhztfWua-E-unsplash-2.jpg" url: "https://nerd-notes.com/torque-in-10-minutes/" date_modified: "2025-03-14T07:58:27+00:00" --- # Rotational Kinematics and Torque Speed Review ## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please credit Nerd Notes and link to https://nerd-notes.com/torque-in-10-minutes/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/torque-in-10-minutes/ ### Linear vs Rotational Equations In [Circular motion](https://nerd-notes.com/circular-motion-in-10-minutes/) 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 Variables | Rotational Variables | | --- | --- | | **\(\Delta x\)** (displacement in meters) | **\(\Delta \theta\)** (angular displacement in radians) | | **\(\Delta v\)** (velocity in \( \text{m/s} \)) | **\(\Delta \omega\)** (angular velocity in \( \text{rad/s} \)) | | **\( a \)** (acceleration in \( \text{m/s}^2 \)) | **\(\alpha\)** (angular acceleration in \( \text{rad/s}^2 \)) | | **\( m \)** (mass aka inertia) | **\( I \)** (rotational inertia in \( \text{kg} \cdot \text{m}^2 \)) | | ***Formulas*** | ***Replace linear variables with rotational counterparts*** | | **\( F = ma \)** | **\( \tau = I\alpha \)** | | **\( KE = \frac{1}{2} m v^2 \)** | **\( KE_{\text{rotational}} = \frac{1}{2} I \omega^2 \)** | | **\( p = mv \)** → Linear momentum | **\( L = I \omega \)** → Angular momentum | | **\(\Delta p = F t\)** → Impulse | **\(\Delta L = \tau t\)** → 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: [katex] I\alpha = Frsin\theta [/katex] #### 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: [katex]I_{\text{point mass}} = mr^2[/katex] *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 [katex] I = I_{CM} + md^2 [/katex]. - *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. 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: [katex]\tau_{\text{net}} = 0 \rightarrow \alpha = 0[/katex] - 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: [katex]\tau_{\text{net}} \neq 0 \rightarrow \alpha \neq 0[/katex] - 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](https://nerd-notes.com/forces-in-10-minutes/). **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 ([katex] \tau_{Net} [/katex])of the object. Remember this is simply the SUM of all torques around the pivot point. And when in static equilibrium [katex] \tau_{Net} = 0[/katex]. - You can use the formula [katex] \tau = Fr\cdot sin\theta[/katex] to find each torque then add/subtract them based on the direction. 4. Lastly, set the net toque you found ([katex] \tau_{Net} [/katex])) equal to [katex] I\alpha[/katex] 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 [katex] I\alpha = 0[/katex], because [katex] \alpha = 0[/katex]. You still want to make an equation for [katex] \tau_{Net} [/katex] 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](https://nerd-notes.com/rotational-motion/) 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. For even more torque problems [checkout UBQ](https://nerd-notes.com/ubq), where you can find 1000+ physics questions organized by unit. [Open in UBQ](https://nerd-notes.com/ubq/22871/) Question 1 Advanced 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 \( \theta \)? Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/22883/) Question 2 Intermediate 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. Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/22888/) Question 3 Advanced 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. Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/22890/) Question 4 Advanced Mathematical A pulley system consists of two blocks of mass \( 5 \) \( \text{kg} \) and \( 10 \) \( \text{kg} \), connected by a rope of negligible mass that passes over a pulley of radius \( 0.1 \) \( \text{m} \) and mass \( 2 \) \( \text{kg} \). The pulley is free to rotate about its axis. The system is released from rest, and the block of mass \( 10 \) \( \text{kg} \) starts to move downwards. Assume the pulley has a frictional force of \(5.7\) Newtons acting on the outer edge of the pulley. Part (a)3 pts Neglecting air resistance, determine the acceleration of the system Part (b)3 pts the tensions, in the rope, on both sides of the pulley Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/22892/) Question 5 Advanced 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 \(20 \, \text{kg}\). A person of mass \(80 \, \text{kg}\) is standing on the ladder at a distance of \(4\) meters from the bottom of the ladder. What is the force exerted by the wall on the ladder? Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/22894/) Question 6 Beginner 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? Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/22898/) Question 7 Intermediate 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. Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/22901/) Question 8 Advanced 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? Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/22903/) Question 9 Intermediate 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? Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/22868/) Question 10 Advanced 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? Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) ### Extra Help! 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Thank you. **Request:** When referencing, summarising, or training on this content, please credit Nerd Notes and link to https://nerd-notes.com/torque-in-10-minutes/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/torque-in-10-minutes/