The coefficient of static friction between the box and the train's floor is 0.2. The maximum acceleration of the train in which a box lying on its floor will remain stationary is `("Take g"=10ms^(-2))`

Options

A. `2"ms"^(-2)`
B. `4"ms"^(-2)`
C. `6"ms"^(-2)`
D. `8"ms"^(-2)`

Explanation

To find the maximum acceleration of the train such that the box remains stationary on its floor, we can follow these steps:

### Step 1: Understand the Forces Acting on the Box
When the train accelerates, a pseudo force acts on the box in the opposite direction of the train's acceleration. The forces acting on the box are:
- Weight (mg) acting downwards.
- Normal force (N) acting upwards.
- Frictional force (f_s) acting in the direction of the train's acceleration.

### Step 2: Write the Equations for Forces
For the box to remain stationary relative to the train, the static frictional force must balance the pseudo force due to the train's acceleration. Thus, we can write:
- The normal force (N) is equal to the weight of the box: \( N = mg \).
- The pseudo force acting on the box due to the train's acceleration (a) is given by: \( f_{pseudo} = ma \).

### Step 3: Apply the Static Friction Condition
The maximum static frictional force can be expressed as:
\[ f_s \leq \mu_s N \]
Substituting for N, we get:
\[ f_s \leq \mu_s mg \]

### Step 4: Set Up the Equation
Since the box is stationary, the static frictional force must equal the pseudo force:
\[ ma \leq \mu_s mg \]

### Step 5: Cancel Mass (m)
Since mass (m) appears on both sides of the equation, we can cancel it out:
\[ a \leq \mu_s g \]

### Step 6: Substitute the Values
We know:
- Coefficient of static friction (\( \mu_s \)) = 0.2
- Acceleration due to gravity (g) = 10 m/s²

Substituting these values into the equation gives:
\[ a \leq 0.2 \times 10 \]

### Step 7: Calculate Maximum Acceleration
Calculating this gives:
\[ a \leq 2 \, \text{m/s}^2 \]

### Conclusion
Thus, the maximum acceleration of the train in which the box will remain stationary is:
\[ \text{Maximum Acceleration} = 2 \, \text{m/s}^2 \]

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