Compute the time period for the following system if the block of mass $m$ is slightly displaced vertically down from its equilibrium position and then released. Assume that the pulley is light and smooth, strings and springs are light. Case (a): the pulley is fixed rigidly to the ceiling, with the spring of constant $k$ connected over the pulley to the mass $m$. Case (b): the pulley itself is a movable pulley hanging in the loop of the string, with the spring of constant $k$ fixed at one end.

Compute the time period for the following system if the block of mass m is slightly displaced vertically down

IV. Exercises · Q6

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Concept understanding — Spring Constant of Combined Springs

Spring Constant of Combined Springs

When you first meet springs, you learn that each spring has a spring constant $k$ — a measure of how stiff it is. A larger $k$ means a stiffer spring: you need more force to stretch or compress it by a given distance. Hooke's law says $F = -k x$ , where $x$ is the displacement from the natural length.

Now imagine you have two springs and you connect them together. What is the stiffness of the combination? That depends entirely on how you connect them.

The Intuition First

Parallel connection — side by side, sharing the load.

Think of two springs placed next to each other, both attached to the same weight. When you pull the weight down, both springs stretch by the same amount. But each spring contributes its own force. So the total force is the sum of the two individual forces. Since force is $k x$ , and $x$ is the same for both, the total force is $(k_1 + k_2) x$ . That means the combination behaves like a single spring with constant $k_1 + k_2$ .

Tip

Parallel springs share the stretch equally. The effective stiffness is the sum: $k_{\text{eff}} = k_1 + k_2 + \dots$

Series connection — end to end, one after the other.

Now imagine two springs attached end to end, hanging from a ceiling with a weight at the bottom. The weight pulls down on the bottom spring, which pulls down on the top spring. The same force passes through both springs. But each spring stretches by a different amount, depending on its own stiffness. The total stretch is the sum of the individual stretches. Since stretch $x = F/k$ , the total stretch is $F/k_1 + F/k_2 = F(1/k_1 + 1/k_2)$ . For the combination, we want $F = k_{\text{eff}} \cdot (\text{total stretch})$ , so $k_{\text{eff}} = 1/(1/k_1 + 1/k_2)$ .

Watch out

A common mistake: thinking series springs add like parallel ones. They don't — series makes the combination softer (smaller $k$ ) than either spring alone.

The Precise Statement

For $n$ springs with spring constants $k_1, k_2, \dots, k_n$ :

In parallel (same displacement, forces add):

k_{\text{eff}} = k_1 + k_2 + \dots + k_n

In series (same force, displacements add):

\frac{1}{k_{\text{eff}}} = \frac{1}{k_1} + \frac{1}{k_2} + \dots + \frac{1}{k_n}

\text{Parallel: } k_{\text{eff}} = \sum k_i \qquad \text{Series: } \frac{1}{k_{\text{eff}}} = \sum \frac{1}{k_i}

Why This Matters …

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Step 1. Case (a): fixed pulley. The pulley is rigidly fixed to the ceiling, and the spring (constant $k$ ) connects over the pulley to the mass $m$ via a light, inextensible string. When the mass is displaced down by $y$ , the string (being inextensible) forces the spring to stretch by exactly the same amount $y$ .

Step 2. The tension in the string equals the spring's restoring force, $T=ky$ , and this same tension acts directly on the mass (since the string is light and the pulley smooth, tension is uniform throughout). So the net restoring force on the mass is $F=-ky$ , giving the standard result $m\ddot y=-ky$ , so $T_{\text{period}}=2\pi\sqrt{m/k}$ -- identical to a mass hanging directly from the spring.

Step 3. Case (b): movable pulley. Here the pulley itself is free to move (hanging in the loop of a string whose other end is fixed, with the spring anchored at one end). When the mass displaces by $y$ , the movable pulley (and hence the loop of string around it) also displaces by $y$ , but because the mass hangs from a string looped around a movable pulley, the spring at the other end stretches by TWICE as much, $2y$ (a standard mechanical-advantage relation for a string looped around a movable pulley). …

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