Rotation its description and what causes it?Consider a disk rotating at constant angular velocity.

Rotation involves turning. Turning implies change of angle.Turning is about an axis of rotation.

All points in the discrotate at the sameangular velocity.

The linear velocity of any part of the discdepends on its distancefrom the center.

��sr

or equivalently s�r � linear velocity v�r�For a compact disc player, the angular velocity of disk

varies as the scan moves inwards – it increases

���� f��i

If this occurs in a time interval � t

then the angular speed is :����

� t

units of angle: 1 radian�3602�

�57.3 degrees

Conversion: radian = ��

180�� (degrees)

Counter clockwise

Clockwise

rotation axisRight Hand Rule

Angular speed = (change in angle)/(time) radians/sec

Angular velocity = directed angular speed.

Angular speed is denoted by ����

� t radians /sec

Angular velocity is denoted by � Direction of � is given by the Right Hand Rule(RHR) If the rate of rotation or � is changing with time

then there is angular acceleration: ��� �

� tradians �s2

Angular acceleration can arise if (1) magnitude of �changes or (2) direction of � changes or(3) both direction and magnitude of � change.

If a rotating object performs 20 revolutions per second thenits angular speed is

omega or ��202��125.6 radians second

or in degrees it is ��8453degreessecond

To set an object into rotation from its non-rotating state is accomplishedby the application of a Torque, which is the analog of Force for rotational motion.

Applying a torque causes angular acceleration. The relation betweenapplied torque and resulting angular acceleration is :

Torque = , angular acceleration = �

Newton's II law for rotational motion is : �I �

I is called the moment of inertia. For a given applied torque, largerthe value of I, smaller is the resulting angular acceleration.

Just as in linear motion, for rotational motion if the torque =0 thenangular acceleration must be =0 too. If angular acceleration is zero then angular speed is a constant.

Also note that the vector angular acceleration points in the same direction as the vector torque.

See page 47 for statement for Newton's second law of motion forrotational motion.

An object rotating about the z axis as shown. Allpoints in the object rotatewith same angular speed.

The linear speed of an elementof mass at a distance r is

v = � r m/s

Vectorially this is v�� x r

�

What is a Torque? and how is it related to applied Force and the axisof rotation?Torque is = (Lever Arm) x (Applied Force)

If you apply a force pointing towards the axis of rotation it willproduce no rotation. If you apply it at right angles to the linejoining the rotation axis to the Force the torque is largest.

Lever arm

The angle that F makes with r is � Lever arm d = r sin� Torque �d F�r F sin�

If ��0 then �0If ��90 then �r F� maximum

Suppose the Force is 100 N and r is 0.2 m then the torque is �20 Nm The torque on the nut is the same but the Force is much larger If the point at which the wrench applies the torque is 0.01 m from the rotation axis, then 20 Nm�F nut �N� x 0.01 m or F nut�2000 N

The mechanical advantage obtained is 0.2 m.01 m

�20�2000100

So a torque wrench is a simple machine

The expression for torque in terms of force and how and where it is applied is

Newton's II law for rotational motion �rF�I �Now ��� ��� t As I is property of the body we can write: ���� I ���� t

Angular Momentum is defined by : L�I �

Newton's II law becomes: ��L� t

for rotational motion

For linear motion it is: F��p� t

What are the units of torque ? Newton meters = Joules

Now using �I �we find the units of I

units of �m xkg msec2

�Iradsec2

Hence I�kg m2� Mass x L2

Moment of Inertia depends on the distribution of mass with respect to the rotation axis.

For linear motion if F is a constant the linear momentum Pis constant

Similarly, for rotational motion if �constantthen the angular momentum L is a constant

The equations of motion are similar: Linear Motion Rotational motion

Initial position x0�������������������Initial angle�0

Initial velociy vx0��������������Initial angular velocity�0

velocity as function of time angular velocity with time given by vx �t ��vx0�at���������angular velocity��t ���0�� t position as function of time angular position with time

given by x �t ��x0�v0 t�1t

a t2����������t ���0��0 t�12� t2

Moment of Inertia of a ring of total mass M and a radius R:

For any small element of mass dm the linear velocity is v�r�

Linear acceleration due to rotation is a�r��

� t���r��

� t

The kinetic energy of an element of mass is :12

m v2�12

m r2�2

For the whole ring the kinetic energy is :12

M R2�2�12

I �2

Hence I�M R2 for the ring. If the mass is distributed at different radii, like for a disc rather than a ring Of same mass M and Radius R the answer for I is different. Shown in next slide

What is the quantity I? An example

Three objects rolling down an incline plane. Which object willreach the ground first. They all have the same mass.

Answer one with the smallestmoment of inertia.

Angular AccelerationA single torque appliedto the spool.

Torque = mg R

�

Assume T1 = T2 and answer the questionwhich direction will the spool turn?

Net torque = T 2 R2�T 1 R1�T �R2�R1�

where T�T 1�T 2

Torque in this position is Mg L/2. As it turns the torque decreases. When it is vertical there is no torque, but it has rotational energy.

If there is no friction in the ball bearings it will oscillate in the vertical plane.

Where will it have the largest angular velocity?

F pN

F fr

Parallel F fr�� F p

Perpendicular N�� F v

Thus Ftot�0

What about torques? Force of gravity goes through the axis of rotation - no torque Normal reaction goes through the axis of rotation - no torqueONLY Torque is due to force of friction It is Counter clockwise about the axis and rolls the wheel down The torque is �F fr x R

Rolling without sliding

Inclined PlaneF�Mg

Only external force is the Force of gravity. Point of contact at rest.

Angular Momentum L�r x p�mr x p

Which direction is the momentum ?

Which direction will be the angular momentum of the skater who is holdingon to the fixed bar?

Will she rotate clockwise or anticlockwise?

What is the linear momentum?

p�mv

What is the angular Momentum?

L�r x p�mvr in magnitude Its direction by RHR is Out of the board Towards you Rotation is CCW

Rotating Bowling ball

Torques about O, the pivot

yellow kid; torque y�md g �L2�cos� CW

blue kid; torque b�m f g �L2�cos� CCW

Net Torque = b� y CCW

Gyroscope

Conservation of Angular Momentum: Change in Momentum of InertiaIncrease in angular velocity.