### 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 Variables | Rotational 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/s^{2}) | 𝛂 (angular acceleration in rad/s^{2}) |

m (mass aka inertia) | I (rotational inertia in kg·m^{2}) |

Formulas | Replace linear variable with rotational counterparts |

F = ma | τ = I𝜶 |

KE = 1/2mv^{2} | KE _{rotational}= 1/2Iω^{2} |

p = mv → Linear momentum | L = Iω → Angular momentum |

∆p = Ft → Impulse | ∆L = τ(t) → Angular impulse |

### 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/s^{2} 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 .

*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: \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

#### 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 **all** torque problem:

- Draw an FBD/diagram of the situation. You will want to also show the lever arm in the diagram.
- 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. - 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.

- 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.

- 0
1.5

2

3

4

g/L

3g/4L

(m+1)g/L

(3mg)/(2L)

None of these.

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