- DEFINITIONS
- MOMENT OF INERTIA
- WORK DONE BY A COUPLE
- ANGULAR MOMENTUM
- COMPARISON OF LINEAR AND ROTATIONAL MOTION

**The number ‘pi’**

Many centuries ago it was discovered that if the circumference
of *any* circle were divided by its diameter, the same number was always
obtained, 3.142 (to 3 decimal places).

**Angles in radians**

**Angular velocity **(or angular speed)

**Frequency, period, speed and angular velocity**

Frequency, f = number of revolution per second ( in hertz, Hz). If, for example, the period T = 0.2 seconds then there are 5 revolutions each second, so we see that:

**Example**

The turntable of a record player makes 45 revolutions a minute. Calculate:

- the angle in radians turned in a minute
- the angular velocity in radians per second
- the linear velocity of a point 15 cm from the centre

**Centripetal force and acceleration**

Consider a cork on the end of a string spun at a constant speed on a circular path:

We can infer that a force is required to keep the cork moving at a constant speed along a circular path because:

- the direction of motion of the cork is constantly changing, which means that
- its velocity is constantly changing, which means that
- it is constantly accelerating, and
- a force is required to produce the acceleration (since F = ma)

**Example**

A 200g mass is swung in a circle of one metre diameter at a uniform speed. It makes 5 revolutions in 10 sec. Calculate:

- its linear and angular speed
- the centripetal acceleration (using both equations above for 'a')
- the centripetal force (using both equations above for 'F')

**Looping the loop**

Consider a mass m spun in a vertical circle of radius r at a constant speed v. To stay on the circle, the mass needs a centripetal force (= mv

- As the speed is reduced, mv
^{2}/r gets smaller, and so does T (mg is constant) - At a particular value of speed, tension T = 0, and
mv
^{2}/r = mg - If the speed gets any less, then mg becomes greater than
mv
^{2}/r, and the mass falls

Calculate the minimum value of speed for an object to be swung
in a vertical circle of radius 1.5 m (g = 10ms^{-2}).

The minimum value of v occurs when:

Notice that the speed does not depend on the mass.

**Rounding a bend**

When a vehicle rounds a bend on a flat road, the centripetal
force required is provided by *friction* between the tyres and the road.
The friction available on a flat corner determines how fast the corner can be
‘taken’.

Cornering that does not depend on friction can be achieved by ‘banking’ a road. Suppose that a car of mass m is moving on a circular path of radius r with a constant speed v:

Suppose that we want the horizontal component of N to provide the necessary centripetal force.

Notice that the angle is independent of the mass.

**Example**

A circular race track has a diameter of 300m. At what angle
should the track be banked for a car to go round it at 100kmh^{-1}, assuming that friction does not
contribute to the centripetal force. (use g = 10ms^{-2}).

**Angular acceleration**

**MOMENT OF INERTIA
**(return
to start of page)

No matter what shape the object is, all the

Compare this with the linear case: KE = ½ m v^{2}. This
suggests that I is equivalent to mass, m. Now
the mass of a body is a measure of its linear inertia, i.e. its 'built in'
opposition to changes in its linear motion, so we infer that:

*The physical significance of the moment of inertia of a body is that it is a measure of the body’s ‘built in’ opposition to changes in its rotational motion*

Two cylinders of different materials, but of equal mass and equal radius are released side by side on an inclined runway. The solid one gets to the bottom first. Why? Consider the definition of moment of inertia:

The hollow cylinder has its mass concentrated further from the
centre (on average), and so the hollow one has a higher value of I, and so a greater ‘built in opposition to changes
in its rotational motion’, and so its rotational acceleration is less than that
of the solid cylinder.

**Example**

Calculate I for the following arrangement of masses about the axis O.

Note: Though I is a ‘rotational property’, its value does not depend upon the state of rotation of an object:*The moment of inertia of an object depends upon the sizes of the masses that make up the object and their distribution about a particular axis*

On the right hand side there are 6.023*10^{23} terms
added together. To get an idea of what that represents, suppose that a computer
could determine and add together a million terms per second. It would then take
the computer 6.023*10^{23}/10^{6} = 6.023 x 10^{17}
seconds to determine all of the right hand side of the above equation.

1.9 x 10^{10} years = 19 000 million years, which is
longer than the age of the Universe (since the Big Bang) which is estimated to
be about 12,000 million years ago (according the announcement by Astronomers in
May 1999).

Thus, we could not possibly, in general, expect to determine values of I directly from the definition. However, for certain objects, particularly symmetric ones, we can determine expressions for I.

**A disc (or solid cylinder) of mass M and radius r**

**WORK DONE BY A
COUPLE **(return
to start of page)

A ‘couple’, such as represented above, consists of two equal and opposite forces whose lines of action do not coincide. A couple tends to produce rotation.

Suppose that the above couple rotates the disc through an angle as represented below:

Notice that:

Thus, we see that**Work and kinetic energy**

**Example**

For the last diagram, if:

i) F= 2N and r = 0.5 m and the disc makes 2 revolution, calculate the work done by the couple.

ii) The mass of the disc is 2kg:

- calculate the moment of inertia of the disc (I =
mr
^{2}/2) - calculate the final angular velocity if the initial angular velocity was zero
- how many revolutions per minute (‘rpm’) does the final angular velocity correspond to?

**ANGULAR MOMENTUM **(return
to start of page)

**a) Derivation**

We consider an object rotating about an axis through O, the axis being at right angles to the page:

By definition, for a revolving particle:

Notice that:

**c) Principle of conservation of angular momentum**

*The total angular momentum of a system remains constant provided that no external torques act on the system*

So, for example, a high-diver decreases his/her moment of inertia by curling up, resulting in his/her angular velocity increasing. (similarly a skater spins faster by folding his/her arms)

**© **

A Level Physics - Copyright © A
C Haynes 1999 & 2004