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# 7.2 Torque and Angular Acceleration

#torque

#force

#rotationalmotion

#nettorque

#momentofinertia

written by peter apps

June 8, 2020

## Enduring Understanding 3.F

A force exerted on an object can cause a torque on that object.

## Essential Knowledge 3.F.1

Only the force component perpendicular to the line connecting the axis of rotation and the point of application of the force results in a torque about that axis.

## Essential Knowledge 3.F.2

The presence of a net torque along any axis will cause a rigid system to change its rotational motion or an object to change its rotational motion about that axis.

## Essential Knowledge 3.F.3

A torque exerted on an object can change the angular momentum of an object.

### Torque

Torque is the rotational equivalent to a Force. A net torque applied on an object will cause the object to change its rotational motion.  The torque depends on 3 factors:

1. The amount of force applied to the object

2. The distance from the pivot point that the force is applied

3. The angle between the radius vector and the force vector. Only the perpendicular part of the force can exert a torque Image courtesy of Hyperphysics.

The torque on an object can be calculated by using the equation: Where 𝜏 represents the torque, F the applied force, r the distance from the pivot and 𝜙 is the angle between the r and F vectors.

If all the torques on an object are balanced (∑𝜏=0) then the object is in equilibrium.

Example: An object weighing 120 N is set on a rigid beam of negligible mass at a distance of 3 m from a pivot, as shown above. A vertical force is to be applied to the other end of the beam a distance of 4 m from the pivot to keep the beam at rest and horizontal. What is the magnitude F of the force required? STEP 1: Identify all the forces in the diagram that could cause a torque

• Weight of the package →  Yes because the force is perpendicular and is 3m from the pivot. This torque will make the beam rotate CCW.

• Applied Force → Yes because it is perpendicular and is 4m from the pivot. This torque will make the beam rotate CW.

STEP 2: Since the beam must remain at rest and horizontal, the two torques must cancel each other out. Setup the equation, substitute in knowns and solve for F. ## Moment of Inertia (I)

The object’s angular acceleration depends on the Moment of Inertia (I). The moment of inertia is the rotational equivalent of mass and describes how difficult it is to rotate an object. For AP 1, you will not need to derive moments of inertia for complex objects, although for AP C you will need to. For a list of common moments of inertia check out this link. For simple objects.  The moment of inertia can be derived using the equation: In general, the closer the masses are to the pivot point, the easier it is to rotate the object. In the example below the when the skater’s arms are outstretched, they have a larger moment of inertia. When they bring their arms inwards, their moment of inertia decreases and it is easier to rotate. Image courtesy of ScienceABC.

### Unbalanced Torques

If there is an unbalanced torque (∑𝜏 ≠ 0) then the object will change its rotational motion. The angular acceleration can be found using the equation: EXAMPLE: A uniform rod of length L and mass M is attached at one end to a frictionless pivot and is free to rotate. The rod is released from rest in the horizontal position. What is the initial angular acceleration of the rod? The moment of inertial of a rod rotated about its end is ⅓ ML^2.

STEP 1: Identify all the forces in the diagram that could cause a torque

• Weight of the rod →  Yes because the force is perpendicular and is 3m from the pivot. Notice that the force acts in the center of the rod, not on the end.

STEP 2: Plug known values into the torque equation.  