TORQUE AND ANGULAR MOMENTUM

1. Torque

1.0  Torque

Definition: A Torque is defined by the force F applied at the distance r from the pivot point. It is the “turning push” that makes something rotate. Think of opening a door, pushing near the hinge is hard, but pushing at the far edge is easy. The effect depends on the followings;

1. how hard you push (force  ),
2. how far from the pivot (distance  ),
3. and the angle of your push.

Quantitatively, torque about a point is given as
where   is the angle between the force and the line from the pivot to the point of application.
Units: newton‑meter (N·m).

Calculations Examples:

1.      A force of 50 N is applied perpendicular to a spanner 0.4 m long. Calculate the torque.

2.      A person pushes a door with a force of 25N at a point 0.8 m from the hinges. The force is perpendicular to the door. Find the turning moment

3.      A force of 60 N acts on a wrench 0.5 m long at an angle of 30° to the handle. Calculate the torque.

4.      A mechanic needs to produce a torque of 120 N·m using a spanner 0.6 m long. The force is applied at 90°. Find the force required.

5.      A force of 80 N produces a torque of 32 N·m. Find the perpendicular distance from the pivot.

2. Angular momentum & Inertia

Angular momentum is the “quantity of rotation” of an object. For a spinning object with moment of inertia   and angular velocity  )
The bigger   or  , the more angular momentum; the object “wants” to keep spinning.

If you recall in the previous studies on linear momentum, you will understand that Torque changes angular momentum (just like force changes linear momentum). Furthermore, if no external torque, angular momentum does not change 

Conservation of angular momentum

The law of conservation of angular momentum says if the net external torque on a system is zero, then its total angular momentum remains constant.

This means that the total angular momentum is constant and that the product of moment of inertia and angular velocity remains constant.

Mathematically:
where   is moment of inertia and   is angular velocity.

Moment of inertia

Moment of inertia   is the rotational mass. it measures how hard it is to change an object’s spin.

Examples

1.     1.  A solid disc has a mass of 6 kg and a radius of 0.40 m. If it rotates at an angular velocity of 15 rad/s, calculate: (a) The moment of inertia of the disc. (b) The angular momentum of the disc.

2.     2.  A flywheel of moment of inertia 10 kg·m² rotates at 12 rad/s. Calculate: (a) Its angular momentum. (b) The new angular velocity if its angular momentum increases to 180 kg·m²/s.

3.      3. A particle of mass 4 kg moves with a speed of 8 m/s in a circular path of radius 3 m. Calculate its angular momentum about the center of the circle.

4.      4. A uniform rod of mass 5 kg and length 2.0 m rotates about an axis passing through its center and perpendicular to its length at 10 rad/s. Calculate: (a) The moment of inertia of the rod. (b) Its angular momentum.

5.      5. A solid sphere of mass 12 kg and radius 0.25 m rotates at an angular velocity of 30 rad/s. Calculate: (a) The moment of inertia of the sphere. (b) The angular momentum of the sphere.

Exercise

Read on Gyroscope and Precession?