2. Simple Harmonic Motion

What is Amplitude? (Definition)

The magnitude of the maximum displacement of the particle from its equilibrium position.

What is Period (T)? (Definition)

The time taken for the body to complete one oscillation.

What is Frequency (f)? (Definition)

The number of oscillations per unit time.

What is the relationship between Period (T) and Frequency (f)?

T = 1/f.

What is the S.I. unit for Period (T)?

seconds (s).

What is the S.I. unit for Frequency (f)?

Hertz (Hz) or s⁻¹.

What is Phase (angle, φ)? (Definition)

The fraction of an oscillation cycle a body is at relative to its starting displacement.

What is Phase Angle for a complete cycle?

2π radians.

What is Angular Frequency (ω)? (Definition)

The rate of change of phase angle of the oscillation, equal to the product of 2π and its frequency f.

What is the S.I. unit for Angular Frequency (ω)?

rad s⁻¹.

What is the relationship between Angular Frequency (ω) and Period (T)?

ω = 2π/T = 2πf.

What is Phase Difference (Δφ)? (Definition)

The difference in the positions of two oscillating bodies in their cycles. It measures how "out of step" two oscillating bodies are.

How is Phase and Phase Difference typically measured?

In degrees or radians.

What is the Defining Equation for Simple Harmonic Motion (S.H.M.)?

a = -ω²x.

What are the key conditions implied by the Defining Equation of S.H.M.?

Acceleration (a) is proportional to displacement (x) from equilibrium, and acceleration is always directed towards the equilibrium position.

What is the direction of acceleration and displacement in S.H.M.?

Always opposite.

Is acceleration constant in S.H.M.?

No, it is directly proportional to displacement, so it varies.

Can kinematics equations be used to analyze S.H.M.?

No, because acceleration is not constant.

What are the characteristics of particle motion in S.H.M.?

Symmetrical about equilibrium, amplitude |OA|=|OB|=x₀, period T is time for one complete oscillation, acceleration always towards equilibrium, maximum acceleration at extreme points (A/B) and zero at equilibrium (O), maximum speed at equilibrium (O) and zero at extreme points (A/B).

What is the Displacement-Time Relationship in S.H.M. (if x=+x₀ at t=0)?

x = x₀ cos(ωt).

What is the Displacement-Time Relationship in S.H.M. (if x=0 at t=0 and moving towards B)?

x = x₀ sin(ωt).

What is the Velocity of N as a function of time (if x=+x₀ at t=0)?

v = dx/dt = -ωx₀ sin(ωt).

What is the Acceleration of N as a function of time (if x=+x₀ at t=0)?

a = dv/dt = -ω²x₀ cos(ωt).

What is the Velocity as a function of displacement in S.H.M.?

v = ±ω√(x₀² - x²).

What is the Acceleration as a function of displacement in S.H.M.? (Defining Equation)

a = -ω²x.

What is the Maximum Velocity in S.H.M.?

v₀ = ±ωx₀ (at equilibrium position).

What is the Maximum Acceleration in S.H.M.?

a₀ = ±ω²x₀ (at amplitude positions).

What is a Horizontal Mass-Spring System? (Model of S.H.M.)

A block of mass m attached to a light spring of force constant k, moving on a horizontal frictionless surface.

What is the Restoring Force in a Horizontal Mass-Spring System?

F_restoring = -kx.

What is the acceleration in a Horizontal Mass-Spring System?

a = -k/m x.

What is the Angular Frequency (ω) in a Horizontal Mass-Spring System?

ω = √(k/m).

What is the Period (T) in a Horizontal Mass-Spring System?

T = 2π√(m/k).

What is a Vertical Mass-Spring System? (Model of S.H.M.)

A spring suspended vertically from the ceiling with a block of mass m hung from its end.

What is the restoring force in a Vertical Mass-Spring System when displaced downwards?

F_restoring = -ky.

What is the acceleration in a Vertical Mass-Spring System?

a = -k/m y.

What is the Angular Frequency (ω) in a Vertical Mass-Spring System?

ω = √(k/m).

What is the Period (T) in a Vertical Mass-Spring System?

T = 2π√(m/k).

What does the Period of a mass-spring system depend on?

Only on the mass of the block and the force constant of the spring.

What is a Simple Pendulum? (Model of S.H.M.)

A bob suspended by a light string.

What is the restoring force in a Simple Pendulum (for small angles)?

F_restoring ≈ -mg sinθ ≈ -mg (s/L).

What is the acceleration in a Simple Pendulum?

a = -g/L s.

What is the Angular Frequency (ω) in a Simple Pendulum?

ω = √(g/L).

What is the Period (T) in a Simple Pendulum?

T = 2π√(L/g).

What does the motion of a simple pendulum system depend on?

Only on the length of the pendulum and the acceleration of free fall (for small angles).

What is the relationship between Uniform Circular Motion and S.H.M.? (Appendix)

The projection of uniform circular motion onto a diameter is Simple Harmonic Motion.

What is the amplitude of S.H.M. from circular motion projection?

Equal to the radius of circular motion.