Question

In the given fig. if M = 12 kg and m = 10kg. Find the acceleration of mass "m" if coefficient of friction = $\frac{11}{12}$ and pulley is an ideal pulley.

Answer 1

0

Answer 2

To find the acceleration of mass 'm', we first need to determine if the system will move. This involves comparing the driving force with the maximum static friction force.

1. Identify the forces involved:

• Driving force: The force that tends to pull the system into motion is the weight of the hanging mass 'm'. $F_{driving} = mg$
• Opposing force: The force that opposes the motion of mass 'M' on the horizontal surface is the static friction force. The maximum static friction force is given by: $f_{s,max} = \mu_s N$ Where $N$ is the normal force acting on mass 'M'. Since mass 'M' is on a horizontal surface and there is no vertical acceleration, the normal force $N$ balances its weight $Mg$. $N = Mg$ Therefore, the maximum static friction force is: $f_{s,max} = \mu_s Mg$

2. Substitute the given values:

• Mass $m = 10 \text{ kg}$

• Mass $M = 12 \text{ kg}$

• Coefficient of friction $\mu = \frac{11}{12}$ (Assuming this is the static coefficient of friction, $\mu_s$).

• Let $g$ be the acceleration due to gravity.

• Calculate the driving force: $F_{driving} = 10 \text{ kg} \times g = 10g$

• Calculate the maximum static friction force: $f_{s,max} = \frac{11}{12} \times 12 \text{ kg} \times g = 11g$

3. Compare the driving force with the maximum static friction force:

• We compare $F_{driving}$ with $f_{s,max}$.
• $10g$ (driving force) vs $11g$ (maximum static friction force)
• Since $10g < 11g$, the driving force is less than the maximum static friction force that the surface can provide.

4. Conclusion:

• Because the pulling force is not sufficient to overcome the static friction, the system will remain at rest.
• Therefore, the acceleration of mass "m" (and mass "M") is zero.

The dimensions (4L and 2L) and the fact that the pulley is ideal are additional information but do not affect the calculation of acceleration in this specific scenario.