Oscillations and SHM: Edge Cases and Subtle Traps (5)

Question

A block of mass $m = 2 \text{ kg}$ is attached to two identical springs (each of stiffness $k = 200 \text{ N/m}$) on a frictionless horizontal surface. The springs are connected on opposite sides of the block to fixed walls and are at their natural length when the block is at rest. Find the period of small oscillations. Then find it again if both springs are connected on the same side in series.

Solution — Step by Step

Final answer: $T_A \approx \mathbf{0.444 \text{ s}}$, $T_B \approx \mathbf{0.889 \text{ s}}$ (twice $T_A$).

Why This Works

The trap is the word “two springs”. Identical springs in different geometries give very different effective stiffness:

• Parallel (both push back when displaced): $k_{eff} = k_1 + k_2$
• Series (one feeds into the next): $1/k_{eff} = 1/k_1 + 1/k_2$

In our problem, “opposite sides of the block” gives parallel even though the springs aren’t physically next to each other — both contribute restoring force simultaneously when the block moves.

Alternative Method

For the parallel case, using energy: PE at displacement $x$ is $\tfrac{1}{2}kx^2 + \tfrac{1}{2}kx^2 = kx^2$. Comparing to standard SHM $U = \tfrac{1}{2}k_{eff}x^2$ gives $k_{eff} = 2k$ — same answer with one less line of algebra.

Common Mistake