Question

An astronaut wants to find out his mass while in orbit, to find out if he is staying healthy while in space. Since he can't use a bathroom scale (why not?), he attaches himself to a spring (k=2500 N/m), pulls himself back from the spring's equil length by 5 m, and times one oscillation to take 1 s.

a.) What is the mass of the astronaut?
________ kg

b.) Find the potential energy stored in the spring when it is furthest from its equilibrium length.
____________J

c.) Find the speed of the astronaut when he passes through the spring's equilibrium.
____________m/s

d.) If he started at a displacement from equilibrium of 6 m instead, how long would one oscillation take now?
____________s

Answer 1

Given that :

spring constant, k = 2500 N/m

spring compression distance, x = 5 m

time-period for oscillations, T = 1 sec

(a) The mass of the astronaut which will be given as :

using an equation,   T = 2m / k                                    

squaring on both sides, we get

T² = 4² m / k

Or   m = T² k / 4²                                                             { eq.1 }

inserting the values in eq.1,

m = (1 s)² (2500 N/m) / 4 (3.14)²

m = [(2500) / (39.4)] kg

m = 63.4 kg

(b) The potential energy stored in the spring when it is furthest from its equilibrium length which will be given as :

using an equation,   P.E_spring = (1/2) k x²                                                                { eq.2 }

inserting the values in eq.2,

P.E_spring = (0.5) (2500 N/m) (5 m)²

P.E_spring = 31250 J

(c) Speed of the astronaut when he passes through the spring's equilibrium which is given as :

using conservation of energy, we have

K.E = P.E_spring

(1/2) m v² = (31250 J)

v² = 2 (31250 J) / m                                                                 { eq.3 }

inserting the value of 'm' in eq.3,

v = (62500 J) / (63.4 kg)

v = 985.8 m²/s²

v = 31.4 m/s

(d) If he started at a displacement from equilibrium of 6 m instead, then time taken for an oscillations which is given as :

t = 1 sec