The period of a piston in an engine, moving simple harmonically is 2 s. A body of mass 10 kg is placed on the piston. Calculate the maximum amplitude of the piston such that the mass is not thrown out from the platform.

Explanation

To solve the problem, we need to determine the maximum amplitude of the piston such that the mass placed on it does not get thrown off. We will follow these steps:

### Step 1: Understand the relationship between period and angular frequency
The period \( T \) of the piston is given as 2 seconds. The angular frequency \( \omega \) can be calculated using the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting \( T = 2 \) seconds:
\[ \omega = \frac{2\pi}{2} = \pi \, \text{rad/s} \]

**Hint:** Remember that the angular frequency is related to the period by the formula \( \omega = \frac{2\pi}{T} \).

### Step 2: Relate acceleration to amplitude
In simple harmonic motion (SHM), the maximum acceleration \( a \) is given by:
\[ a = \omega^2 A \]
where \( A \) is the amplitude.

**Hint:** The maximum acceleration in SHM can be expressed in terms of angular frequency and amplitude.

### Step 3: Set the condition for the mass not being thrown off
For the mass to not be thrown off the piston, the maximum acceleration must be equal to the acceleration due to gravity \( g \):
\[ a = g \]
Substituting the expression for acceleration from SHM:
\[ \omega^2 A = g \]

**Hint:** The condition for the mass to remain on the piston is that the maximum acceleration must not exceed gravitational acceleration.

### Step 4: Solve for amplitude \( A \)
Now, substituting \( \omega = \pi \):
\[ (\pi)^2 A = g \]
Thus, we can express \( A \) as:
\[ A = \frac{g}{\pi^2} \]

**Hint:** Rearranging the equation will allow you to isolate the amplitude.

### Step 5: Substitute the value of \( g \)
Using \( g \approx 9.81 \, \text{m/s}^2 \):
\[ A = \frac{9.81}{\pi^2} \]
Calculating \( \pi^2 \approx 9.87 \):
\[ A \approx \frac{9.81}{9.87} \approx 0.993 \, \text{m} \]

**Hint:** Use the approximate value of \( \pi \) for calculations to find the numerical value of amplitude.

### Final Answer
The maximum amplitude of the piston such that the mass is not thrown out from the platform is approximately:
\[ A \approx 0.993 \, \text{m} \]