Simple Harmonic Motion I

1. A toy rifle employs a spring whose force constant is 200 N/m. In use, the spring is compressed 50 mm, and when released, it propels a 5g rubber ball. What is the ball's speed when it leaves the rifle ?

   Energy conservation

   ½mv² = ½kA²

   v = A(k/m)^½ = 5 x 10^-2 (200/5 x 10^-3)^½ = 10 m/s

   
   
   
2. A force of 5 lb compresses a spring by 3.5 inches.


• a)What is the elastic PE of the compressed spring ?

k = F/x = 5/(3.5/12) = 17.1 lbs/ft

PE = ½kx² = ½ x 17.1 x (3.5/12)² = 0.73 ft.lbs



• b)A 0.5 lb ball is placed against the compressed spring. When the spring is let go, what is the ball's initial speed ?



Energy conservation. Using the result from above question

v = A(k/m)^½ = (3.5/12) x (17.1/(0.5/32))^½ = 9.6 ft/s




3. A spring whose force constant is 15 lb/ft oscillates with a period of 0.6 s when a bag of onions is suspended from it. What is the weight of the onions ?



T = 2π(m/k)^1/2

m = (T²k)/4π² = (0.6 x 0.6 x 15)/4π² = 0.14 slugs

weight = 0.14 x 32 lbs = 4.48 lbs



4. A spring has a 1.0 s period of oscillation when a 20N weight is suspended from it. Find the elongation of the spring when a 50 N weight is suspended from it.



T = 2π(m/k)^1/2

k = 4π²m/T² = 4π²w/T²g

x = F/k = (F T²g)/(4π²w) = (50 x 1 x 9.8)/(4π² x 20) = 0.62 m