01-Simple Harmonic Motion

Question 1

Definition of Simple Harmonic Motion (SHM)

Answer 1

Oscillatory motion in which acceleration is directly proportional to the displacement and always directed towards the equilibrium position.

Question 1

In SMH, in which direction is the body accelerating?

Answer 1

Towards the centre of the motion (except at the centre of the motion where the acceleration is zero).

Question 2

Time Period & Frequency of the oscillation:

Answer 2

-A restoring force tries to return system to equilibrium.
-The system has inertia and overshoots equilibrium position
-The object oscillates with simple harmonic motion because its acceleration is proportional to the displacement from equilibrium and always acts towards equilibrium .

Question 3

What is Damping?

Answer 3

The dissipation of energy over time

Question 4

Types of Damping

Answer 4

1. Light damping: amplitude of oscillation is reduced gradually.
2. Critical Damping: returns to its equilibrium position in the shortest possible.
3. Heavy Damping: returns to the equilibrium position very slowly.

Question 5

Simple Harmonic Motion (SHM) formula

Answer 5

𝑎 𝛼−𝑥, where a ≡ acceleration; x ≡ displacement
𝑎= − 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 × 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑥.

Question 6

Time Period & Frequency of the oscillation formula

Answer 6

𝜔=2𝜋/𝑇 𝑜𝑟 2𝜋𝑓
T=1/𝑓, 𝑓=1/T

Question 7

Mathematical expression for displacement in SHM formula

Answer 7

-𝑎=−𝜔^2 𝑥
-𝜔= 2𝜋𝑓
-If x =+A when t =0 at zero velocity: 𝑥 =𝐴 cos⁡( 𝜔t)
-If the value of x is 0 when t =0: 𝑥 =𝐴 𝑠𝑖 𝑛⁡( 𝜔t)

Question 8

Time Period in a mass-spring system formula

Answer 8

-Hook’s Law, 𝐹 = 𝑘𝑥
-Newton’s 2nd Law, F= ma, 𝑎 = −(𝑟𝑒𝑠𝑡𝑜𝑟𝑖𝑛𝑔 𝑓𝑜𝑟𝑐𝑒)/𝑚𝑎𝑠𝑠= (−𝑘𝑥)/𝑚
-Combining those two equations, we get 〖𝑚𝜔〗^2 𝑥=𝑘𝑥; 𝑚𝜔^2=𝑘; 𝜔=√(𝑘/𝑚)
2𝜋/𝑇=√(𝑘/𝑚), 𝑇=2𝜋√(𝑚/𝑘)

Question 9

Time Period in a pendulum formula

Answer 9

-𝑚𝑔𝑐𝑜𝑠𝜃 perpendicular to the path or the bob.
-𝑚𝑔𝑠𝑖𝑛𝜃 along the path towards the equilibrium position.
-The restoring force, 𝐹=−𝑚𝑔𝑠𝑖𝑛𝜃
-Acceleration, 𝑎=𝐹/𝑚 =−𝑚𝑔sin𝜃/𝑚 = −𝑔𝑠𝑖𝑛𝜃
-θ does not exceed approximately 10°, 𝑠𝑖𝑛𝜃=𝑠/𝐿 and 𝑎 =−𝑔/𝐿 𝑠=−𝜔^2 𝑠, 𝑤ℎ𝑒𝑟𝑒 𝜔^2=𝑔/𝐿
-2𝜋/𝑇=√(𝑔/𝐿), 𝑇=2𝜋√(𝐿/𝑔)

Question 10

The energy equation in SHM formula

Answer 10

-The potential energy Ep changes with displacement x from equilibrium, in accordance with the equation
-𝐸𝑝 =1/2 𝑘𝑥^2
-𝐸T=1/2 𝑘𝐴^2
-𝐸𝑇=𝐸𝐾+𝐸𝑃
-𝐸𝑘=𝐸𝑇−𝐸𝑃=1/2 𝑘 (𝐴^2−𝑥^2 )