Question

A block rides on a piston that is moving vertically with simple harmonic motion. The maximum speed of the piston is $2\,{\text{m}}{{\text{s}}^{{\text{ - 1}}}}$. At what amplitude of motion will the block and piston separate? ($g = 10\,{\text{m}}{{\text{s}}^{{\text{ - 2}}}}$)
A. $20\,{\text{cm}}$
B. $30\,{\text{cm}}$
C. $40\,{\text{cm}}$
D. $50\,{\text{cm}}$

Answer 1

We are asked to calculate the amplitude of motion when the block and piston separate. First recall the formula for maximum velocity and use it to form an equation. Then recall the condition required for the block and piston to separate and apply it to find the required answer.

Complete step by step answer:
Given, a block rides on a piston that is moving vertically with simple harmonic motion.
Maximum speed of the piston, $v = 2\,{\text{m}}{{\text{s}}^{{\text{ - 1}}}}$
We are asked to find the amplitude of motion when the block and the piston separate.
The maximum velocity is given by the formula,
${v_{\max }} = A\omega$ (i)
where $A$ is the amplitude and $\omega$ is the angular frequency.
Here maximum velocity is $2\,{\text{m}}{{\text{s}}^{{\text{ - 1}}}}$, putting this value in equation (i) we get,
$2 = A\omega$
$\Rightarrow A\omega = 2$ (ii)
At the moment when the block and piston separates the restoring force on the block will be equal to the weight of the block.Let $m$ be the mass of the block.Then the weight of the block can be written as,
$W = mg$ (iii)
The restoring force is given by the formula,
$F = m{\omega ^2}A$ (iv)
Now, equating the weight of the block with the restoring force we get (equating equation (iii)and (iv)),
$m{\omega ^2}A = mg$
$\Rightarrow {\omega ^2}A = g$
$\Rightarrow \omega A\omega = g$
Substituting the value of $A\omega$ from equation (ii) we get,
$\omega \times 2 = g$
Putting the value of $g$ we get,
$\omega \times 2 = 10$
$\Rightarrow \omega = \dfrac{{10}}{2} = 5\,{\text{rad}}\,{{\text{s}}^{{\text{ - 1}}}}$
Putting this value of $\omega$ in equation (ii) we get,
$A \times 5 = 2$
$\Rightarrow A = \dfrac{2}{5}$
$\Rightarrow A = 0.4\,{\text{m}}$
The amplitude of the motion when the block and the piston separate is $0.4\,{\text{m}}$.
The options are given in centimetre so we convert the unit to centimetre and we get the amplitude as,

\Rightarrow A = 0.4 \times 100\,{\text{cm}} \\\ \therefore A = 40\,{\text{cm}}$$ **Hence, the correct answer is option C.** **Note:** Simple harmonic motion is a type of motion where the restoring force is proportional to the object’s displacement from its mean position and acts towards the mean position. In simple harmonic motion, the object moves forward and backward from its mean position or equilibrium position.