Figure 4.22 shows a massless spring fixed at the bottom end...

Figure $4.22$ shows a massless spring fixed at the bottom end of an inclined of inclination $37^{\circ}\left(\tan 37^{\circ}=\right.$ $3 / 4$ ). A small block of mass $2 \mathrm{~kg}$ start slipping down the incline from a point $4.8 \mathrm{~m}$ away from free end of spring. The block compresses the spring by $20 \mathrm{~cm}$, stops momentarily and then rebounds through a distance $1 \mathrm{~m}$ up the inclined, then $\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)$. (A) Coefficient of friction between block and inclined is $0.5$. (B) Coefficient of friction between block and inclined is $0.75$. (C) Value of spring constant is $1000 \mathrm{~N} / \mathrm{m}$. 69 (D) Value of spring constant is $2000 \mathrm{~N} / \mathrm{m}$.

Figure $4.22$ shows a massless spring fixed at the bottom end of an inclined of inclination $37^{\circ}\left(\tan 37^{\circ}=\right.$ $3 / 4$ ). A small block of mass $2 \mathrm{~kg}$ start slipping down the incline from a point $4.8 \mathrm{~m}$ away from free end of spring. The block compresses the spring by $20 \mathrm{~cm}$, stops momentarily and then rebounds through a distance $1 \mathrm{~m}$ up the inclined, then $\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)$.
(A) Coefficient of friction between block and inclined is $0.5$.
(B) Coefficient of friction between block and inclined is $0.75$.
(C) Value of spring constant is $1000 \mathrm{~N} / \mathrm{m}$.
69
(D) Value of spring constant is $2000 \mathrm{~N} / \mathrm{m}$.

Key Concepts

Conservation of Energy

This concept involves accounting for all forms of energy in a system as they transform from one form to another. In mechanical systems, gravitational potential energy, kinetic energy, and elastic potential energy (in the case of a spring) are interconverted. The work done against non-conservative forces, such as friction, must also be considered to accurately analyze the motion and energy changes in the system.

Work Done Against Friction

Work done against friction is the energy dissipated due to the frictional force acting over a distance. It is calculated as the product of the frictional force and the displacement along the path. This energy loss is critical in problems where sliding motion is involved, as it affects the net energy available for converting into other forms and directly influences the motion.

Motion on an Inclined Plane

Analyzing motion on an inclined plane requires the decomposition of forces into components parallel and perpendicular to the plane. The gravitational force is split into components that contribute to the acceleration along the plane and those that are balanced by the normal force. This breakdown is essential for understanding the dynamics of objects moving on inclines, particularly when friction and additional forces, such as spring forces, are present.

Hooke's Law and Spring Compression

Hooke's Law describes the linear relationship between the force exerted by a spring and its displacement from equilibrium, expressed as F = kx. The elastic potential energy stored in a spring is given by (1/2)kx^2, where k is the spring constant and x is the displacement. This relationship is fundamental in problems involving spring compression or extension, especially when energy conversion between kinetic, gravitational, and elastic potential energy is analyzed.

Please give Ace some feedback