Question

A $35$-cm-long vertical spring has one end fixed on the floor. Placing a $2.2$ kg physics textbook on the spring compresses it to a length of $29$ cm. What is the spring constant?

Short answer

The spring constant$k=360$N/m

Step-by-step solution

Step 1. Given Information

A vertical spring with $35$cm length and a $2.2$kg of physics textbook compressed it to a length of$29$cm

Step 2. Hooke's Law

When a spring is compressed, a force is exerted by its material to restore it to its equilibrium position and this force is called restoring force $F_{sp}$. Hooke's law state the relationship between the restoring force and the vertical displacement $∆y$

$F_{sp}=-k∆y$

where $k$is the force constant and the negative sign shows the direction

Step 3. Find the spring constant

The book presses against the free end of the spring, so it exerts a force in the negative sign of the restoring due to its weight $mg$, so the force is given by

$$\begin{gathered} F=F_{sp} \\ mg=-k∆y \\ k=-\frac{mg}{∆y} \end{gathered}$$

the length of the spring changes from $35$cm to $29$cm, we put the values

$$\begin{gathered} k=-\frac{mg}{∆y} \\ =\frac{(2.2kg)(9.8m/s^2)}{0.29m-0.35m} \\ =360N/m \end{gathered}$$