Answer :
The astronaut's mass is found by comparing the square of the periods before and after changes in mass. The new period of oscillation was used along with the known mass and period to solve for the new mass, which is approximately 51.2 kg.
The astronaut's oscillation on a chair attached to springs is related to simple harmonic motion. The period of oscillation depends on the mass (m) and the spring constant (k) as given by the formula T = 2π sqrt(m/k). Since the spring constant hasn't changed, we can compare the periods with the respective masses to find the new mass of the astronaut.
Firstly, we establish the relationship between period and mass for the known case of 55.0 kg, which yielded a period of 2.85 s:
T1 = 2π sqrt(m1/k)
For the second measurement with the new unknown mass (m2) and period of 2.75s:
T2 = 2π sqrt(m2/k)
By taking the ratio of the squares of the two periods, we eliminate the spring constant:
(T2/T1)2 = m2/m1
We can solve for m2:
m2 = m1 × (T2/T1)2
Inserting the known values:
m2 = 55.0 kg × (2.75/2.85)2
Calculating this gives us:
m2 = 55.0 kg × 0.9307 = 51.2 kg (approximately)
So, the astronaut's mass now is approximately 51.2 kg, which corresponds to option (f).
