Simple Harmonic Motion (SHM)

SHM

Linear restoring force = ____?

In SHM, the position = sinusoidal function of time

Remember: In SHM, the position = sinusoidal function of time

x(t) = A cos (ωt + ∅)

Equation for position in SHM?

Amplitude (A)

what is A?

Angular frequency (ω) = 2πf = 2π / T

What is ω?

Phase constant (∅)

What is ∅?

Amplitude (A)

• tells you how far the object swings around back and fourth

angular frequency (ω)

• the cycle repeats itself every period

• T = 2π / ω

the phase constant (∅)

• can be positive, 0, or negative

• it shifts the cosine curve

  • when positive: to the left

  • when negative: to the right

to the left

When the cosine curve is positive it gets shifted by the phase constant in what direction?

to the right

When the cosine curve is negative it gets shifted by the phase constant in what direction?

Velocity in SHM

• Eqn: v(t) = -Aωsin(ωt+∅)

• inverted sine function

• sin(ωt+∅) oscillates btw -1 & 1

  • Vmax occurs when sin(ωt+∅) = -1

  • Vmax = Aω

-1

what is sin(ωt+∅) equal to when vmax occurs?

inverted sine function

what function is velocity in SHM?

v(t) = -Aωsin(ωt+∅)

Equation for velocity in SHM?

Vmax = Aω

Equation for Vmax in SHM?

acceleration in SHM

• Eqn: a(t) = -Aω²cos(ωt+∅)

• inverted cosine function

• cos(ωt+∅) oscillates btw -1 & 1

  • amax occurs when cos(ωt+∅) = -1

  • amax = Aω²

-1

what is cos(ωt+∅) equal to when amax occurs?

inverted cosine function

what function is acceleration in SHM?

a(t) = -Aω²cos(ωt+∅)

equation for acceleration in SHM?

amax = Aω²

equation for amax in SHM?

The further the object is from the point of equilibrium, the larger the restoring force, so the larger the accel.

Remember: The further the object is from the point of equilibrium, the larger the restoring force, so the larger the accel.

v = 0 & a = -amax = -Aω²

When x = A (furthest the object can be from point of accel):

the object moves the fastest when it reaches the equilibrium point (x=0)

Remember: the object moves the fastest when it reaches the equilibrium point (x=0)

v = -vmax = -Aω & a = 0

When x = 0:

v = 0 & a = amax = Aω²

when x = -A:

acceleration points in the opposite direction as the displacement (a = inverted cosine, ∆x = cosine)

Remember: acceleration points in the opposite direction as the displacement (a = inverted cosine, ∆x = cosine)

the magnitude of acceleration is proportional to the displacement

Remember: the magnitude of acceleration is proportional to the displacement

a(t) = -w²x(t)

The equation for the magnitude of acceleration in relation to the displacement:

-kx = m(-ω²x)

The magnitude of the linear restoring force:

• the linear restoring force is the net force F=ma

f = 1/2π√k/m

equation for frequency in a mass-spring system where k is the spring constant:

T = 2π√m/k

equation for period in a mass-spring system where k is the spring constant:

frequency and period have nothing to do with the amplitude

Remember: frequency and period have nothing to do with the amplitude

F = 1/2π√g/L

equation for frequency in for a pendulum system where the constant k=mg/Length of pendulum:

T = 2π√L/g

equation for the period in for a pendulum system where the constant k=mg/Length of pendulum:

frequency and period of a pendulum do NOT depend on the mass of the bob

Remember: frequency and period of a pendulum do NOT depend on the mass of the bob