Physics

Problem

6. An object swings on the end of a cord as a simple pendulum with period T. Another object oscillates up and down on the end of a vertical spring also with period T. If the masses of both objects are doubled, what are the new values for the Periods? Pendulum (A)

\frac{T}{\sqrt{2}}

(B) T (C)

T \sqrt{2}

(D)

T \sqrt{2}

Mass on Spring

T \sqrt{2}

T \sqrt{2}

\frac{T}{\sqrt{2}}

(A) Pendulum:

\frac{T}{\sqrt{2}}

, Mass on Spring:

T \sqrt{2}

(B) Pendulum: T, Mass on Spring:

T \sqrt{2}

T \sqrt{2}

, Mass on Spring: T (D) Pendulum:

T \sqrt{2}

, Mass on Spring:

\frac{T}{\sqrt{2}}

Intuitive explanation

The key is recognizing that the period formulas for these two systems depend differently on mass. For a pendulum, $T = 2\pi\sqrt{L/g}$ —mass doesn't appear at all, so doubling it changes nothing. For a spring, $T = 2\pi\sqrt{m/k}$ —mass appears under the square root, so doubling $m$ multiplies the period by $\sqrt{2}$ . Think about why: gravity pulls equally on all masses, but a spring's resistance to acceleration depends on the object's inertia.

Intuitive explanation

Browse the problem library