If a mass of 10 kg is kept on the inclined plane at the end...

If a mass of $10 \mathrm{~kg}$ is kept on the inclined plane at the end of a spring and slowly allowed to come to rest, the spring has a maximum compression of $25 \mathrm{~cm} .$ The friction coefficient between plane and mass is $\mu=\frac{1}{2 \sqrt{3}}$. (i) What is the spring constant $\mathrm{k}$ ? (ii) Now the mass is kept at a farther point $\mathrm{A}$ on the plane and released from rest. The spring has a maximum compression of $2 \mathrm{~m}$. What is the height descended by the mass? $\left(\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$

If a mass of $10 \mathrm{~kg}$ is kept on the inclined plane at the end of a spring and slowly allowed to come to rest, the spring has a maximum compression of $25 \mathrm{~cm} .$ The friction coefficient between plane and mass is $\mu=\frac{1}{2 \sqrt{3}}$.
(i) What is the spring constant $\mathrm{k}$ ?
(ii) Now the mass is kept at a farther point $\mathrm{A}$ on the plane and released from rest. The spring has a maximum compression of $2 \mathrm{~m}$. What is the height descended by the mass? $\left(\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$

Key Concepts

Gravitational Potential Energy on an Incline

Gravitational potential energy depends on the vertical height change of an object, even if the motion occurs along an inclined path. Understanding how to express the change in gravitational energy using the vertical displacement, rather than the distance along the incline, is crucial for energy balance calculations in these problems.

Friction in Mechanical Systems

Friction is a force that resists motion and converts mechanical energy into thermal energy. In analyzing systems on inclined planes, the frictional force—dependent on the coefficient of friction and the normal force—modifies the net work done on the object and must be considered when applying energy conservation methods.

Equilibrium on an Inclined Plane

When an object is at rest on an inclined plane, the forces acting on it—such as gravitational, normal, frictional, and any additional forces like those from a spring—must balance along and perpendicular to the plane. Analyzing this equilibrium is key to relating these forces and solving for unknowns such as the spring constant.

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. This linear relationship allows one to determine the spring constant by measuring the spring's compression or extension when a known force is applied.

Work-Energy Principle

The work-energy principle involves equating the work done by all forces, including non-conservative forces like friction, to the change in the system's mechanical energy. This principle is essential when evaluating how gravitational potential energy is converted into elastic potential energy in a spring and energy lost to friction.

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