13.1.1 simple harmonic motion

Part 1 : Introducing circular motion

Equations of circular motion

It is useful to investigate the equations of circular motion as these are used to derive the equations of simple harmonic motion (SHM)

In atomic physics, space travel and astronomy, there are many examples of bodies moving in circular (or very close) paths

A body travelling equal distances in equal times along a circular path has constant speed but not constant velocity

Speed is scalar has magnitude only Velocity is a vector has magnitude and

direction A body moving in a circular motion is

accelerating – why? At any instant the direction of a body motion is

along the tangent of the circular path (Newton’s first law of motion)

υ1υ2

The time taken for one rotation is called the period, T

The number of rotations in a unit time is the angular frequency, ƒ measured in Hertz

Again T = 1 or ƒ = 1

ƒ T

e.g. what is the angular frequency of the Earth as it rotates on its axis?

Angular displacement; angles in radians

For body moving in a circle it is often useful to state its position in terms of the angle through which it has moved relative to its starting position

This is angular displacement - measured in radians (rad) not degrees

Calculating angles in radians

•Angle = s / r (in radians)

•If s = r then = 1 radian

•1 radian is the angle subtended at the centre of a circle by an arc equal in length to the radius

If s = 2r (circumference of the circle):

= s = 2r = 2 radians = 360r r

1 radian 57

It follows that the length s of and arc subtending an angle at the centre of the circle of radius r is:

s = r

Speed around a circular path

To calculate the speed of a body moving in a circular path, you need to know the distance moved and the time taken i.e.

Speed = circumference of circletime to complete one rotation

Since the circumference of a circle = 2r = 2r

t

Angular velocity, The speed of a body moving in a circle can be

specified by(i) Speed along the tangent at a given instant or(ii) Angular velocity (in rads-1)

Angular velocity is the angle swept out in a given unit of time by the radius joining the body to the centre of the circle

Or the angle in radians swept out by the radius every second

Symbol = (omega)

Consider a body moving in a circular path Radius OA rotates through angle = /t (linear s = d/t) If arc AB has length s, and is the

constant speed of the body, then: = s/t From earlier, s = r Hence = r = r

t

Angular velocity and frequency The angular frequency is the number of

revolutions per second = /t A complete revolution of a circle = 2 rad So = 2/t Since ƒ = 1/T = 2ƒ

Quick check questions

1. Convert the following angles in degrees to radians: 360; 90 ; 60 ; 45

2. Convert these angles in radians to degrees: 1 rad; 0.25 rad; rad; /5 rad

3. An aircraft is circling above an airport. Its path has a diameter of 20km and its speed is 120m/s. How long will it take to complete one circuit of its path? In what time interval will the direction change by 30?

4. Calculate the angular speed of a masonry drill bit rotating at 720 rev/s

5. Calculate the speed of the edge of the tip if the diameter of the bit in question 4 is 6.0 mm

Circular acceleration

When an object moves in a circle, its velocity is at a tangent to the circle

Its velocity is changing since the direction is changing the body is accelerating

shows how the velocity vector changes. The arrow shows the direction of the change in velocity the direction of the acceleration

Acceleration towards the centre of a circle is the centripetal acceleration

An object can only accelerate if a resultant force is acting on it – the centripetal force – otherwise the object would fly off at a tangent to the circle (Newton’s first law of motion)

In the two examples previously, the moving object is acted on by the force towards the centre of the circle, but it does not get any closer

The centripetal force needed to make an object follow a circular path depends on

(i) the mass, m of the object

(ii) the speed, (iii) the radius, r of the circle

Centripetal force, F = m2

r

Centripetal acceleration

Since F = ma; a = F/m a = 2

r Now = r so centripetal acceleration can be

calculated using angular velocity: a = r2

And centripetal force = ma

= mr2

Quick check questions

1. A particle moves in a semicircular path AB of radius 5.0m with constant speed 11m/s. Calculate

(a) The time taken to travel from A to B

(b) The average velocity

(c) The average acceleration

A B

5m5m

2. A turntable makes 33 revolutions per minute. Calculate:

(a) The angular velocity in rad/s

(b) The linear velocity of a point 0.12m from the centre

3. A grinding wheel of diameter 0.12m spins horizontally. P is a typical grinding particle bonded to the edge of the wheel. The rate of rotation is 1200 rev/min, calculate:(a) The angular velocity(b) The acceleration of P(c) The magnitude of the force acting on P if its mass is 1.0 x 10-4 kg

The maximum radial force at which P remains bonded is 2.5N(d) Calculate the angular velocity at which P will leave the wheel if rotation rate is increased(e)if the wheel exceeds this rotation what will the speed and direction of P be immediately after it leaves the wheel?

0.12m

P

Derivation of a = v2/r

Strictly speaking it is more correct to derive centripetal acceleration and then use F = ma to show that F = m2/r

•A body moves at constant speed,

in a circle of radius r

•It travels from A to B in time t

Arc AB = t Since s = r, arc AB = r r = t So = t [1]

r

Let vectors A and B represent velocities at A and B

= B - A or B + (A)

By the parallelogram law

Resultant = velocity = vector represented by XZ

YZ = -A in size () and direction (CA)

XY = B in size () and direction (BD)

Since -A is perpendicular to OA and B is perpendicular to OB:

XYZ = AOB = If t is very small then is very small and

XY will have the same length as arc XZ below:

Because is very small arc XZ straight line

Since s = r arc XZ = From [1] = t

r

XZ = x t

r

XZ = 2t

r

The magnitude of acceleration between A and B is:

a = velocity = XZ

time t Hence a = 2t

rt

a = 2

r

Speed of a body moving in a circle: = r So a = (r)2

r

a = 2r If t is so small that A and B all but

coincide; XZ is perpendicular to A or B i.e. along line AO or BO. Therefore the body has centripetal acceleration