Figure (8-E7) shows a spring fixed at the bottom end of an...

Figure (8-E7) shows a spring fixed at the bottom end of an incline of inclination $37^{\circ} .$ A small block of mass $2 \mathrm{~kg}$ starts slipping down the incline from a point $4 \cdot 8 \mathrm{~m}$ away from the spring. The block compresses the spring by $20 \mathrm{~cm}$, stops momentarily and then rebounds through a distance of $1 \mathrm{~m}$ up the incline. Find (a) the friction coefficient between the plane and the block and (b) the spring constant of the spring. Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$.

Key Concepts

Energy Conservation in Mechanical Systems

This concept involves accounting for the transformation of energy from one form to another within a system. It is crucial when analyzing how gravitational potential energy, kinetic energy, and elastic potential energy interplay while incorporating the work done by non-conservative forces like friction, which dissipates energy as heat.

Inclined Plane Dynamics

This concept is about studying the motion of an object along a sloped surface. It involves breaking down gravitational forces into components parallel and perpendicular to the incline, which is essential for determining acceleration and the normal force, both critical when analyzing friction and motion on slopes.

Friction and Energy Dissipation

Friction is a non-conservative force that opposes motion, converting mechanical energy into thermal energy. Understanding how to characterize friction through the coefficient of friction and incorporating its effects into work-energy calculations is vital in predicting the motion and energy loss of moving bodies on surfaces.

Hooke's Law and Elastic Potential Energy

Hooke's Law states that the force exerted by a spring is proportional to its displacement from equilibrium, with the proportionality constant being the spring constant. The energy stored in a deformed spring is determined by its elastic potential energy, which is essential when analyzing energy conversions in systems that involve spring compression or extension.

Transcript

00:01 In this question, we have a block with mass of 2 kilograms resting on an incline slope, which has an incline of 37 degrees.

00:08 There is a spring fixed at the bottom of this incline.

00:11 When the block is released, it slides down the slope and compresses the spring by a distance of 0 .2 metres.

00:17 What we want to find is the coefficient of friction between this block and the slope, and we want to find the spring constant of the spring.

00:25 So in order to do the first part of this question, we can apply the work energy principle for the down.

00:30 Motion of the body.

00:32 We know that initially all of its energy is stored as gravitational potential energy, and if we take the bottom point to have zero gravitational potential energy, we know that at this point all of the energy is elastic potential energy.

00:45 However, some energy has been lost between the start and final position due to the work done against friction.

00:52 So what we can say is that mg, and we must resolve this component parallel to the slope, so as we can see here, this angle is equal to 37 as well, so this component is sign 37.

01:11 So this is the force, but we must multiply by the distance, which is 4 .8 times 0 .2, to obtain the work done.

01:20 And this must be equal to the work done against friction, where the frictional force is given by muar.

01:26 So again, multiply by the distance, we've got 5 mu .r, plus the elastic potential energy due to the compression of the spring, which is given by 1⁄2.

01:38 X squared...

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