Calculate the maximum acceleration of a moving car so that a body lying on the floor of the car remains statio

Question

Calculate the maximum acceleration of a moving car so that a body lying on the floor of the car remains stationary. The coefficient of static friction between the body and the floor is $0.15$. (take $g=10 	ext{ m s}^{-2}$)

Accepted Answer

1. **Analyze the Physics:** For the body to remain stationary relative to the accelerating car, the force responsible for accelerating the body (static friction) must not exceed the limiting friction. 
2. **Identify Forces:**
   - Downward force: Weight $W = mg$.
   - Upward force: Normal reaction $N = mg$.
   - Horizontal force required for acceleration $a$: $F = ma$.
3. **Apply Friction Condition:** The static friction $f_s$ provides the necessary force to accelerate the body along with the car. The condition for the body not to slip is:
   $$ f_s \le f_{s,max} = \mu_s N $$
   $$ ma \le \mu_s mg $$
   $$ a \le \mu_s g $$
4. **Calculate Maximum Acceleration:**
   $$ a_{max} = \mu_s g $$
   Given $\mu_s = 0.15$ and $g = 10 	ext{ m s}^{-2}$:
   $$ a_{max} = 0.15 	imes 10 = 1.5 	ext{ m s}^{-2} $$
   (Reference: NCERT Class 11, Physics Part I, Chapter 5: Laws of Motion, Section 5.9 and Example 5.7, which solves an identical problem for a box in a train [1]).

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