To determine whether the mass will remain in equilibrium on a 10-degree incline, we need to analyze the forces acting on the mass.

1. Forces Involved

When a mass is placed on an inclined plane, the following forces act on it:

• Gravitational force ($F_g$): The weight of the object, $F_g = mg$, where $m$ is the mass and $g$ is the acceleration due to gravity.
• Normal force ($F_n$): The force perpendicular to the surface of the incline.
• Component of gravitational force parallel to the incline ($F_{g,\parallel}$): This is the portion of the gravitational force causing the object to potentially slide down the incline, given by $F_{g,\parallel} = mg \sin \theta$, where $\theta = 10^\circ$.
• Frictional force ($F_f$): The force that resists the sliding motion, calculated using $F_f = \mu F_n$, where $\mu$ is the coefficient of friction and $F_n = mg \cos \theta$.

2. Equilibrium Condition

For the mass to be in equilibrium, the net force parallel to the incline must be zero, meaning the frictional force must balance the parallel component of the gravitational force: $F_f \geq F_{g,\parallel}$

That is: $\mu mg \cos \theta \geq mg \sin \theta$

Simplifying: $\mu \geq \tan \theta$

Substituting $\theta = 10^\circ$: $\mu \geq \tan 10^\circ \approx 0.176$

Conclusion

• If the coefficient of friction $\mu$ is greater than or equal to 0.176, the frictional force will be enough to keep the mass in equilibrium.
• If $\mu < 0.176$, the mass will slide down the incline because the gravitational component parallel to the incline will exceed the available friction.

Follow-up

Do you want to know more details about the forces, or have specific numbers to analyze?

Expanding Questions

1. How does the mass of the object affect the forces acting on the incline?
2. What would happen if the incline angle was increased beyond 10 degrees?
3. How would adding a horizontal force affect the equilibrium of the mass?
4. What role does the normal force play in determining equilibrium on an incline?
5. How does the coefficient of friction influence whether an object stays at rest or slides?

Tip:

Always calculate the frictional force to ensure the object remains in equilibrium on an incline.