A mass is hung from a spring. Which is longer the period of vertical oscillations or horizontal swings.?

Question

The question also adds that the amount dfference must also be specified. There is also a hint to allow the unstretched spring to be L1 and the stretched spring with the mass to be L2.

Answer 1

If we assume that both the vertical oscillations and the horizontal swings are harmonic motion, then:

The period of the horizontal swings is given by:

T(h) = 2πsqrt(L2/g), where L2 is the length of the stretched spring with mass m, and g is the gravitational acceleration

The period of the vertical oscillation is given by:

T(v) = 2πsqrt(m/k), where m is the mass attached to the spring and k is the spring constant

But since F = kx = k(L2-L1) = mg

This means that (m/k) = (L2-L1)/g

Substitute the above into our equation for the period of vertical oscillation:

T(v) = 2πsqrt(m/k) = 2πsqrt((L2-L1)/g)

This clearly shows that T(h) > T(v), the peiod of the horizontal swings is greater than the period of the vertical oscillations

ΔT = T(h) - T(v) = 2πsqrt(L2/g) - 2πsqrt((L2-L1)/g)

= 2πsqrt(L2/g) - 2πsqrt(L2/g)sqrt(1-(L1/L2))

= 2πsqrt(L2/g) (1 - sqrt(1-(L1/L2)))