© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Understand the scenario of circular motion and SHM projection
A point A moves uniformly around a circle of radius $0.36\,\text{m}$. In $0.1\,\text{s}$, it covers an angle of $30^\circ$. When we project A perpendicularly onto the diameter MN, this projection, labeled P, executes Simple Harmonic Motion (SHM). We want to find the restoration force per unit mass (i.e., the maximum acceleration) when P is at the extremity M.
Step 2: Determine the angular speed of A
The point A covers $30^\circ$ (which is $\frac{\pi}{6}$ radians) in $0.1\,\text{s}$.
Thus, the angular speed $ \omega $ is:
$ \omega \;=\;\frac{\text{angular displacement}}{\text{time}}
\;=\;\frac{\frac{\pi}{6}}{0.1}
\;=\;\frac{\pi}{6}\,\times\,10
\;=\;\frac{5\pi}{3}\,\text{rad/s}. $
Step 3: Relate angular speed to the linear velocity
For uniform circular motion of radius $R = 0.36\,\text{m}$, the linear speed $v$ is:
$ v \;=\; R\,\omega \;=\; 0.36\,\times\,\frac{5\pi}{3}
\;=\;0.36\times\frac{5\pi}{3}
\;=\;0.6\pi\,\text{m/s}. $
Numerically, $ 0.6\pi \approx 1.884\,\text{m/s}. $
Step 4: Find the expression for acceleration in SHM
In SHM, the displacement of the projection P from the central position is $ x = R \sin(\omega t) $. The acceleration is given by
$ a \;=\; -\,\omega^2\,x. $
The magnitude of the acceleration is maximum at the extremity ($|x| = R$), so the maximum acceleration is:
$ a_{\max} = \omega^2 \,R. $
Step 5: Calculate the maximum acceleration and hence the force per unit mass
We have
$ \omega = \frac{5\pi}{3} \quad \text{and} \quad R = 0.36\,\text{m}. $
Therefore,
$ \omega^2 = \left(\frac{5\pi}{3}\right)^2 \;=\;\frac{25\,\pi^2}{9},
\quad a_{\max} = \omega^2\,R = \frac{25\,\pi^2}{9} \times 0.36. $
First multiply $25 \times 0.36 = 9$ to get
$ a_{\max} = \frac{9\,\pi^2}{9} \;=\;\pi^2 \approx 9.87\,\text{m/s}^2. $
Force per unit mass (which is acceleration) at the extremity is therefore approximately $9.87\,\text{N/kg}.$
Step 6: State the final answer
Hence, the restoration force per unit mass when P touches M is about $9.87\,\text{N}.$
