1.1. To investigate the relationship between To investigate the relationship between circular motion and SHMcircular motion and SHM

2.2. To be able to calculate the displacement at To be able to calculate the displacement at any particular moment under different any particular moment under different starting conditionsstarting conditions

3.3. To be able to calculate the velocity at any To be able to calculate the velocity at any particular moment.particular moment.

Book Reference : Pages 38-39Book Reference : Pages 38-39

x

y

r

r cos r sin Consider an object in Consider an object in circular motion... At any circular motion... At any given time the x and y given time the x and y coordinates are given by:coordinates are given by:

x = r cos x = r cos y = r sin y = r sin

Graphically the x coordinate changes as shownGraphically the x coordinate changes as shownx

+r

-r

/ radsinewavedisc.swsinewavedisc.swff

We have seen that the displacement of an object in SHM We have seen that the displacement of an object in SHM is described by a is described by a sinusoidalsinusoidal function. However, there are function. However, there are differences between where we start the motion at time differences between where we start the motion at time t=0t=0

x+A

-A

x +A

-A

Maximum Maximum displacement A displacement A at time t=0 at time t=0 (cosine)(cosine)

Zero Zero displacement at displacement at time t=0 (sin)time t=0 (sin)

Providing that we Providing that we operate in radians,operate in radians, the the displacement displacement xx of an object in SHM is given by: of an object in SHM is given by:

x = A cos 2x = A cos 2ft ft (for displacement A at time t=0)(for displacement A at time t=0)

x = A sin 2x = A sin 2ft ft (for zero displacement at time t=0)(for zero displacement at time t=0)

Where A is the maximum displacementWhere A is the maximum displacement

Calculator must be in radians for this to workCalculator must be in radians for this to work

The velocity of an object in SHM is given by:The velocity of an object in SHM is given by:

v = v = 2 2f f (A(A22 – x – x22))

Where v is the velocity, f is the frequency, A is the maximum Where v is the velocity, f is the frequency, A is the maximum displacement and x is the displacementdisplacement and x is the displacement

The velocity is considered positive if moving away The velocity is considered positive if moving away from the equilibrium point and negative if moving from the equilibrium point and negative if moving towards ittowards it

Note this will be a maximum when x=0Note this will be a maximum when x=0vvmax max = 2= 2fAfA

In a similar way the maximum value for In a similar way the maximum value for acceleration given by the acceleration equation acceleration given by the acceleration equation we saw last lesson will be....we saw last lesson will be....

Acceleration = - (2f)2 x displacement

Accelerationmax = - (2f)2 A

Where f is the frequency, and A is the maximum displacement .Where f is the frequency, and A is the maximum displacement .

A spring oscillates in SHM with a period of 3s and an A spring oscillates in SHM with a period of 3s and an amplitude of 58mm. Calculate:amplitude of 58mm. Calculate:

a.a. The frequency The frequency [0.33 Hz][0.33 Hz]b.b. The maximum acceleration The maximum acceleration [0.25 ms[0.25 ms-2-2]]

The displacement of an object oscillating in SHM The displacement of an object oscillating in SHM changes with time and is described by changes with time and is described by

X (mm) = 12 cos 10t where t is the time in seconds after X (mm) = 12 cos 10t where t is the time in seconds after the object’s displacement was at its maximum value. the object’s displacement was at its maximum value. Determine:Determine:

a.a. The amplitude The amplitude [12mm][12mm]b.b. The time period The time period [0.63s][0.63s]c.c. The displacement at t = 0.1s The displacement at t = 0.1s [6.5mm][6.5mm]

An object on a spring oscillating in SHM has a time An object on a spring oscillating in SHM has a time period of 0.48s and a maximum acceleration of period of 0.48s and a maximum acceleration of 9.8m s9.8m s-2-2. Calculate:. Calculate:

a.a. Its frequency Its frequency [2.1 Hz][2.1 Hz]

b.b. Its Amplitude Its Amplitude [0.057m][0.057m]

An object oscillates in SHM with an amplitude of An object oscillates in SHM with an amplitude of 12mm and a period of 0.27s. Calculate12mm and a period of 0.27s. Calculate

a.a. The frequency The frequency [3.7 Hz][3.7 Hz]

b.b. Its displacement and direction of motion 0.1s, Its displacement and direction of motion 0.1s, and 0.2s after the displacement was +12mmand 0.2s after the displacement was +12mm

[-8.2mm towards maximum negative displacement][-8.2mm towards maximum negative displacement][-0.7mm towards maximum positive displacement][-0.7mm towards maximum positive displacement]