AP Physics C Simple Harmonic Motion/Oscillations

When is an object in simple harmonic motion?

If the acceleration of the object is proportional to the object’s displacement from an equilibrium position and that acceleration is directed towards the equilibrium position (a∝Δx)

acceleration is opposite of the direction of the displacement from equilibrium position (eg. going towards)

What is amplitude A?

Maximum displacement from equilibrium position

What is w for a mass-spring system?

Angular frequency

w = sqrt(k/m) = Δθ/θt = 2π/T

What is the condition for SHM?

d²x/dt² = w²x

dt2d2θ​=−ω2θ​​ need to memorize!

What is the period of a mass spring system?

T = 2πsqrt(M/K)

Are frequency and angular frequency the same thing?

No,

frequency f is the number of cycles an object goes through per second and the frequency is equal to the inverse of the period (f = 1/T)

Angular frequency is equal to 2pi/T so it is equal to 2pif

What is position as a function of time in SHM?

x(t) = Acos(wt + Φ) must use radians

What is phase constant/phase shift?

shifts the location of the cosine/sine wave on the horizontal axis

What is velocity as a function of time in SHM?

v(t) = -Awsin(wt +Φ)

can derive by taking the derivative of x(t) for SHM

What is the maximum velocity of an object in SHM?

from this equation, v(t) = -Awsin(wt +Φ) , and knowing range of sine is -1 to 1, the maximum velocity is equal to v = Aw

What is acceleration as a function of time in SHM?

a(t) = -Aw²cos(wt +Φ)

can derive by taking the derivative of v(t) for SHM

What is the maximum acceleratioin of an object in SHM?

from this equation, a(t) = -Aw²cos(wt +Φ) , and knowing range of cosine is -1 to 1, the maximum velocity is equal to a_max= Aw²

What is the amplitude of a mass-string system?

when the block is at its maximum displacement from the equilbrium position

Is simple harmonic motion also uniformly accelerated motion?

nah, bc acceleration changing

Is mechanical energy conserved in simple harmonic motion?

Yes, because no energy is added or removed from the system via a work done by an external force and no energy is dissipiated via a work done by friction so ME is conserved in SHM

When the block is located at its maximum displacement from equilibrium position/the amplitude, what is the mechanical energy?

It has no velocity so no kinetic energy

Therefore ME = EPE which is 1/2kx² which in this case would be ME = 1/2kA²

When the block is located at equilibirum position, what is the mechanical energy?

string isn’t being stretched so therefore no EPE and only KE

therefore ME = 1/2mv²

What is the angular frequency and period for a linear simple harmonic oscillator (SHO)

w = (k/m)^0.5

T = 2π(m/k)^0.5

What are the periods for different types of pendulums?

Torsion pendulum: T = 2π(I/k)^0.5

Simple pendulum: T = 2π(L/g)^0.5

Physical pendulum: T = 2π(I/mgh)^0.5 (unlike simple pendulum, can be any shape pivoting)

What is damped harmonic motion?

The mechanical energy E in a real oscillating system decreases during oscillations because external forces like drag force inhibit oscilation and transfer mechanical energy to thermal energy. Thus, the motion is said to be damped.

If damping force is given to be F_d = -bv where v is the velocity of oscillator and b is the damping constant, then displacement is

x(t) = Ae^-bt/2mcos(w’t+Φ) where w’t = sqrt((k/m)-(b²/4m²⁾⁾

If damping constant is very small, then w’ = w and w’ is just sqrt(k/m)

For a small b, mechanical energy is equal to E(t) = 0.5kx^_m²e^-bt/m

Forced oscillations and resonance?

If an external driving force with angular frequency w_d acts on an oscillating system with natural angular frequency w, the system oscillates with angular frequncy w_d.

The velocity amplitude v_m is greatest when w_d = w (resonance). Amplitude x_m is approximately greatest under the same condition

What is the total potential energy for a system in SHM?

E_total = 0.5kA²