A2-kg mass attached to a spring that is compressed 40 cm has a maximum velocity of 10 m/s. if the spring is then compressed by 50 cm, what is the speed when the spring is at half its amplitude?

10.83 m/s

Explanation:

Case 1:

m = mass attached to spring = 2 kg

x = compression of the spring = 40 cm = 0.40 m

k = spring constant

v₀ = maximum speed = 10 m/s

Using conservation of energy

maximum spring potential energy = maximum kinetic energy

(0.5) k x² = (0.5) m v₀²

k (0.40)² = (2) (10)²

k = 1250 N/m

Case 2 :

m = mass attached to spring = 2 kg

A = amplitude = 50 cm = 0.50 m

x = compression of the spring at half amplitude = A/2 = 50/2 = 25 cm = 0.25 m

k = spring constant = 1250 N/m

v = speed = ?

Using conservation of energy

maximum spring potential energy = spring potential energy + kinetic energy

(0.5) k A² = (0.5) k x² + (0.5) m v²

(1250) (0.50)² = (1250) (0.25)² + (2) v²

v = 10.83 m/s

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