13.1.1 Shm Part 2 Circular To Shm

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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