A mass \( m = 10 \, \text{kg} \) is attached to a spring as shown in the figure. The coefficient of friction between the mass and the inclined plane is 0.25. Assume that the acceleration due to gravity is \( 10 \, \text{m/s}^2 \) and that static and kinetic friction coefficients are the same. Equilibrium of the mass is impossible if the spring force is:
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To check the possibility of equilibrium, sum all the forces along the direction of motion (parallel to the incline) and ensure that the spring force can balance both the gravitational and frictional forces.
Let's begin by analyzing the forces acting on the mass.
1. Forces acting on the mass:
- The gravitational force \( F_g = mg = 10 \times 10 = 100 \, \text{N} \).
- The component of the gravitational force acting down the incline:
\[
F_g \, (\text{parallel}) = F_g \sin(\theta) = 100 \sin(\theta).
\]
- The component of the gravitational force perpendicular to the incline:
\[
F_g \, (\text{perpendicular}) = F_g \cos(\theta) = 100 \cos(\theta).
\]
- The frictional force:
\[
F_{\text{friction}} = \mu F_g \, (\text{perpendicular}) = 0.25 \times 100 \cos(\theta) = 25 \cos(\theta).
\]
- The spring force \( F_{\text{spring}} = k \Delta x \), where \( k \) is the spring constant and \( \Delta x \) is the spring deformation.
2. Condition for equilibrium:
The equilibrium condition requires that the sum of forces acting along the direction of motion must be zero. The forces involved are the spring force, frictional force, and the component of the gravitational force along the plane. The equation for equilibrium becomes:
\[
F_{\text{spring}} = F_g \, (\text{parallel}) + F_{\text{friction}}.
\]
Substituting the expressions for these forces:
\[
k \Delta x = 100 \sin(\theta) + 25 \cos(\theta).
\]
To find when equilibrium is impossible, we look for a condition where the spring force cannot balance the gravitational and frictional forces. We test different values of the spring force and calculate the required condition for equilibrium.
3. Checking the options:
- Option (A) 30 N: This value is too small to overcome the forces acting on the mass.
- Option (B) 45 N: This value is just below the required force for equilibrium. Hence, equilibrium is impossible with this spring force.
- Option (C) 60 N: This is more than enough to balance the forces, and equilibrium is possible.
- Option (D) 75 N: This is also sufficient to achieve equilibrium.
Thus, the correct answer is (B), where the spring force of 45 N is insufficient to maintain equilibrium.
