A 0.500-kg mass suspended from a spring oscillates with a period of 1.18 s. How much mass must be added to the object to change the period to 2.07 s?

The add mass = 5.465 kg

Explanation:

Note: Since the spring is the same, the length and Tension are constant.

f ∝ √ (1/m) ... Equation 1 (length and Tension are constant.)

Where f = frequency, m = mass of the spring.

But f = 1/T ... Equation 2

Substituting Equation 2 into equation 1.

1/T ∝ √ (1/m)

Therefore,

T ∝ √ (m)

Therefore,

T₁/√m₁ = k

where k = Constant of proportionality.

T₁/√m₁ = T₂/√m₂ ... Equation 3

making m₂ the subject of the equation

m₂ = T₂² (m₁) / T₁² ... Equation 4

Where T₁ = initial, m₁ = initial mass, T₂ = final period, m₂ = final mass.

Given: T₁ = 1.18 s m₁ = 0.50 kg, T₂ = 2.07 s.

Substituting into equation 4

m₂ = (2.07) ² (0.5) / (1.18) ²

m₂ = 4.285 (1.392)

m₂ = 5.965 kg.

Added mass = m₂ - m₁

Added mass = 5.965 - 0.5

Added mass = 5.465 kg.

Thus the add mass = 5.465 kg