Mechanics

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).

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:

1. the angle in radians turned in a minute
2. the angular velocity in radians per second
3. 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)
The force F keeping the cork on its circular path acts along the string towards the centre, and is called a centripetal (=‘centre seeking’) force. We can show that:

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:

1. its linear and angular speed
2. the centripetal acceleration (using both equations above for 'a')
3. 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 (= mv2/r). At the top of the motion, this force is partly provided by the force of gravity and partly by the tension T in the string.

• As the speed is reduced, mv2/r gets smaller, and so does T (mg is constant)
• At a particular value of speed, tension T = 0, and mv2/r = mg
• If the speed gets any less, then mg becomes greater than mv2/r, and the mass falls
Example

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 individual particles that make it up move on a circular path at the same angular velocity. For the particle labelled m1:

Compare this with the linear case: KE = ½ m v2. 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
Observation
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
In simple situations such as the above, we can work out the moment of inertia by taking into account every individual particle. However, suppose that we had an object made of just 12 grams of carbon. The number of carbon atoms in 12 g of carbon is defined as Avogadro’s number, and equals 6.023 x 1023. So the object contains 6.023 x 1023 atoms of carbon, and every one of these produces a term in the definition of I:

On the right hand side there are 6.023*1023 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*1023/106 = 6.023 x 1017 seconds to determine all of the right hand side of the above equation.

1.9 x 1010 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 torque in the rotational case corresponds to force in the linear case.

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:

1. calculate the moment of inertia of the disc (I = mr2/2)
2. calculate the final angular velocity if the initial angular velocity was zero
3. 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:

b) Newton’s second law

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)

COMPARISON OF LINEAR AND ROTATIONAL MOTION (return to start of page)