A mass of 0.2 kg is attached to a horizontal spring whose stiffness is 12 N/m. Friction is negligible. At t = 0 the spring has a stretch of 3 cm and the mass has a speed of 0.5 m/s.

a: What is the amplitude (maximum stretch) of the oscillation?

b: What is the maximum speed of the block?

a:

Initial State: s_i = 3 cm, v_i = 0.5 m/s

Final State: v_f = 0 (maximum stretch)

b:

Initial State: s_i = 3 cm, v_i = 0.5 m/s

Final State: Maximum speed (s_f = 0)

System: Mass and spring

Surroundings: Earth, table, wall (neglect friction, air)

a:

From the Energy Principle:

$E_{f} = E_{i} + W$

$K_{f} + U_{f} = K_{i} + U_{i} + W$

$K_{f}$ and $W$ cancel out.

$0 + \dfrac{1}{2}k_{s}s^2_{f} = \dfrac{1}{2}mv^2_{i} + \dfrac{1}{2}k_{s}s^2_{i}$

$\dfrac{1}{2}k_{s}s^2_{f} = \dfrac{1}{2}(0.2 kg)(0.5 m/s)^2 + \dfrac{1}{2}(12 N/m)(0.03 m)^2$

$\dfrac{1}{2}k_{s}s^2_{f} = 0.03 J$

$s_{f} = \sqrt {2(0.03 J)/(12 N/m)} = 0.07 m$

$s_{f} = 0.07 m$

b:

$K_f + 0 = 0.03 J$

$\dfrac{1}{2}mv^2_{f} = 0.03 J$

$v_{max} = \sqrt {2(0.03 J)/(0.2 kg)} = 0.55 m/s$