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Answer the following questions for a mass that is hanging on a spring and oscillating up and down with simple harmonic motion. Note: the oscillation is small enough that the spring stays stretched beyond its rest length the entire time.

Answer top/bottom/top and bottom/equilibrium/nowhere:

**1) Where in the motion is the magnitude of the force from the spring on the mass zero?**

Equilibrium, because the change in position delta x is zero.

**2) Where in the motion is the magnitude of the net force on the mass a maximum?**

The top, because the force of the spring (from compression) and the force of gravity both act on the mass.

**3) Where in the motion is the magnitude of the net force on the mass zero?**

The bottom, because the force of gravity and the force from the spring oppose each other to keep the block at rest (away from the equilibrium position)

**4) Where in the motion is the magnitude of the acceleration a maximum?**

The top, because force is maximum there.

**5) Where in the motion is the speed zero?**

It is zero at the top and bottom; it changes direction at the top and stays at rest at the bottom.

**6) Where in the motion is the acceleration zero?**

The bottom, since the net force is zero.

**7) Where in the motion is the speed a maximum?**

At equilibrium.

**8) Where in the motion is the magnitude of the force from the spring on the mass a maximum?**

The top and bottom, because of delta x.

Yes/No

**1) When the object is at half its amplitude from equilibrium, is its speed half its maximum speed?**

x(t) = Acos(ωt + φ), cos(ωt + φ) = cos(30 deg) = .5

v(t) = -ωAsin(ωt + φ) = (sqrt(3)/2)Aω, maximum v(t) is Aω

No.

**2) When the object is at half its amplitude from equilibrium, is the magnitude of its acceleration at half its maximum value?**

Yes.

a(t) = -(ω^2)Acos(ωt + φ) = .5Aω, maximum a(t) is A(ω^2)