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### Transcript of 13.1.1 Shm Part 2 Circular To Shm

• 1. 13.1.1 simple harmonic motion Part 2: translating circular motion to simple harmonic

2. Simple harmonic motion

• Linear motion: a - constant in size + direction
• Circular motion: a - constant in size only
• Oscillatory motion: a changes periodically in size + direction (likexand )
• SHM is a special form of oscillating motion
• Pendulums and masses on springs exhibit SHM
• A body oscillates with SHM if the displacement changes sinusodially

3. Linking circular motion and SHM

• The arrangement shown below can be used to demonstrate the link between circular motion and SHM

4.

• Adjusting the speed of the turntable will allow both shadows to have the same motion (i.e. move in phase)
• The shadows are the components of each motion parallel to the screen and are sinusoidal
• Let N represent the sinusoidal motion of one shadow
• It oscillates about O (equilibrium point) in a straight line between A and B

O N A B 5. Displacement and velocity

• When N isleftof O:
• -xis left
• -is left when moving away from Oand right when moving towards O
• When N isrightof O:
• -xis right
• -is right when moving away from Oand left when moving towards O

6.

• The size of the restoring force increases withxBUT always acts towards equilibrium point (O)
• F- x
• resulting acceleration must behave likewise, since F = ma and m is constant
• i.e. a increases withxbut acts towards O
• a- x
• In oscillations a andxalways have opposite signs

7. Definition

• If the acceleration of a body isdirectlyproportional to the distance from a fixed point, and is always directed toward that point, then the motion is simple harmonic
• a- xor
• a = -(+ve constant)x
• Many mechanical oscillations are nearly simple harmonic, especially at small amplitudes
• Any system obeying Hookes law will exhibit SHM when vibrating

8. Equations of SHM

• Consider the ball rotating on the turntable
• The ball moves in a circle of radius r
• It has uniform angular velocity
• The speed,varound the circumference will be constant and equal to r( v= r)
• At time t the ball (and hence the bob of the pendulum) are in the positions shown:

9. 10. Displacement

• Angle= t(since= /t)
• The displacement of the ball along OF from O is given by:x= r cos= r cos t
• For the pendulum (and masses on springs), the radius of the circle is equal to the amplitude of its oscillation i.e. r = A
• Hencex= A cos t
• But= 2 f
• x= A cos 2 ft

r x 11. 12. Velocity

• The velocity of the pendulum bob is equal to thecomponentof the balls velocity parallel to the screen (i.e. along y-axis)

Bob Ball v= r O r velocity= - vsin Ball Bob 13.

• Velocity of bob = - vsin= - r sin
• = /t = t
• Velocity of pendulum bob,= - r sin t
• Sinis +ve when 0 180 (i.e. bob or ball moving down)
• Sinis ve when 180 360 (bob or ball moving up)
• Negative sign ensures velocity is negative when moving down and positive when moving up!

14. Variation of velocity with displacement

• sin 2 + cos 2 = 1
• sin = (1- cos 2 )
• = r sin
• = r (1- cos 2 )
• From earlierx= r cos , sox /r = cos
• ( x /r) 2= cos 2

15.

• By substituting:
• Velocity = r (1- cos 2 )
• = r (1 - ( x /r) 2 )
• = (r 2x 2 )
• Recall= 2 f
• Hence velocity of pendulum in SHM of amplitude A is given by:
• =2 f (A 2x 2 )

16. Acceleration

• The acceleration of the bob is equal to thecomponentof the acceleration of the ball parallel to the screen
• The acceleration of the ball,a = 2 r towards O
• So the component of a along OF = 2 r cos

Bob Ball a = 2 /ra = 2 r cos O O 17.

• Hence, the acceleration of the bob is given by:
• a = - 2 r cos (-ve since moving down)
• Sincex= r cosand= 2 f
• a = -(2 f ) 2x
• Since (2 f ) 2 or 2is a +ve constant, equation states that acceleration of the bob towards the equilibrium point O is proportional to the displacementxfrom O
• The acceleration is zero at O and maximum when the bob is at the limits of its motion when the direction and motion changes

18. Time period

• Period T is time taken for the bob to complete one oscillation
• In the same time the ball has made one revolution of the turntable
• T = circumference of circle
• speed of ball
• T = 2 r

19.

• Since= r
• T = 2
• For a particular SHMis constant and independent of the amplitude (or radius) of the oscillation
• If the amplitude increases, the body travels fasterT is unchanged
• A motion with constant T, whatever the amplitude, isisochronous- this is an important characteristic of SHM

20. Time traces of SHM

• Displacement

T/4 T/2 3T/4 T Note: the gradient = velocity 21.

• Velocity

T/4 T/2 3T/4 T Note: when= 0, a = max 22.

• Acceleration

T/4 T/2 3T/4 T a = 0 when= max 23.

• All graphs are sinusoidal
• When the velocity is zero, the acceleration is a maximum and vice versa
• There is aphase differencebetween them
• Betweenand a phase difference = T/4
• Betweenxand a phase difference = T/2

24. Summary: equations of SHM

• Frequencyf= /2
• Period T = 2 /
• Displacementx= A cos t
• = A cos 2 f t
• Velocity = A 2x 2
• =2 f A 2x 2
• Acceleration a = -(2 f ) 2 x