Simple Harmonic Motion

Question 1

SHM definition

Answer 1

is any motion in which the unbalanced force is proportional but in the opposite direction to the displacement from the rest position.

Question 2

F = -ky

Answer 2

F = Force (N)
k = a constant
y = spring extension

Question 3

y = Asinwt - when y = 0 and t = 0

y = Acoswt - when y = A and t = 0

Answer 3

y = displacement (m)
A = amplitude (m)
w = angular velocity (rad s-1)
f = frequency (Hz)

Important when calculating problems involving the sine and cosine of w make sure your calculator is in RADIANS mode.

Question 4

derive a = -wy^2 starting at y = Asinwt

Answer 4

y = Asinwt
dy/dt = (Asinwt)
v = Awcoswt
dv/dt = (Awcoswt)
a = -Aw^2sinwt
since Asinwt = y
then
a = -w^2y

Question 5

derive a = -wy^2 starting at y = Acoswt

Answer 5

y = Acoswt
dy/dt = (Acoswt)
v = - Awsinwt
dv/dt = (Awsinwt)
a = -Aw^2coswt
since Acoswt = y
then
a = -w^2y

Question 6

a = -w^2y

Answer 6

a = acceleration
w = angular velocity (rad s-1)
y = displacement (m)

Question 7

derive v = w√A^2 - y^2 starting at v = Awcoswt

Answer 7

v = Awcoswt
1 - sin^2wt = cos^2wt
√1-sin^2twt = coswt
v = ±Aw√1-sin^2wt
y = Asinwt
y^2 = A^2sin^2wt
v = ±Aw√1 - y^2/A^2
v = ±√A^2 w√1 - y^2/A^2
v = ±w√A^2( 1 - y^2/A^2)
v = ±w√A^2 - y^2

Question 8

Derive Kinetic Energy for SHM

Answer 8

Ek = 1/2mv^2
since v = ±w√A^2 - y^2
Ek = 1/2 m (v = ±w√A^2 - y^2)^2
so
Ek = 1/2mw^2 (A^2 - y^2)

Question 9

Summary of SHM formula

Answer 9

displacement

y = Asinwt - when y = 0 and t = 0
y = Acoswt - when y = A and t = 0

frequency/period

w =2πf
w = 2π/T

velocity

v = Awcoswt
v = ±w√A^2 - y^2
max velocity = wA

acceleration

a = -w^2y
max acceleration = w^2A

kinetic energy

Ek = 1/2mw^2(A^2 - y^2)
max Ek = 1/2mw^2A^2

Potential Energy

Ep = 1/2mw^2y^2

Question 10

Damping

Answer 10

Undamped = normal SHM
Under Damped = amplitude decreases slowly with time
Critical Damping = amplitude to zero in the shortest possible time
Over Damped = does not oscillate but moves slowly till rest

Question 11

v = Awcoswt

Answer 11

v = velocity (ms-1)
A = amplitude (m)
w = angular velocity (rad s-1)
t = time (s)

Question 12

if an object falls off an another oscillating object what is the magnitude and direction of acceleration.

Answer 12

9.8ms-2 DOWNWARDS.

Question 13

how can you create damping

Answer 13

increase the surface area of the mass