A 1 kg block situated on a rough incline is connected to a...

A $1 \mathrm{~kg}$ block situated on a rough incline is connected to a spring of spring constant 100 $\mathrm{N} \mathrm{m}^{-1}$ as shown in Fig. 6.17. The block is released from rest with the spring in the unstretched position. The block moves $10 \mathrm{~cm}$ down the incline before coming to rest. Find the coefficient of friction between the block and the incline. Assume that the spring has a negligible mass and the pulley is frictionless.

Key Concepts

Conservation of Energy with Non-Conservative Work

This concept involves accounting for all energy transfers in a system, including energy stored in conservative fields (like gravitational and elastic potential energy) and energy lost due to non-conservative forces such as friction. It is used to set up an energy balance where the work done by friction is explicitly taken into account.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position within a gravitational field. On an inclined plane, a change in height translates into a change in gravitational potential energy, which can be converted into other forms of energy or work against friction.

Spring Potential Energy

Spring potential energy is the energy stored in a spring when it is stretched or compressed. This energy is given by the formula (1/2)kx², where k is the spring constant and x is the displacement from the equilibrium position. It plays a crucial role in systems where mechanical energy is temporarily stored in spring deformation.

Frictional Forces and the Coefficient of Friction

Frictional forces arise due to the interaction between surfaces in contact and act to oppose relative motion. The coefficient of friction quantifies this resistance and is determined by the nature of the surfaces in contact. In problems involving inclined planes, friction does work over the distance of motion, affecting the overall energy balance of the system.

Transcript

00:03 In this problem, it is saying that a mass block of 1 kg is kept on an inclined plane which has angle of inclination 37 degree.

00:13 Now this box is connected with the help of a string and other end of the string is connected to a spring.

00:21 So initially the spring was unstretched.

00:24 Now this block is released from rest.

00:27 So its initial velocity is 0.

00:30 So here, u is equal to 0 meter per second.

00:34 Now after stretching of 10 cm, it again comes to rest.

00:40 So we have to determine what should be the coefficient of friction for this surface.

00:45 So its final velocity, v is again here 0 meter per second.

00:53 So first of all, let us draw the free body diagram of this block at any intermediate location.

01:00 So what are the forces that will be acting on it? so first one will be the normal reaction that will be acting perpendicular to the surface.

01:09 The second one will be weight of the box that will be acting in downward direction.

01:16 Third one will be the friction force that will be acting in upward direction along the plane.

01:23 So in this direction friction force will be acting.

01:28 Now we can resolve the forces in the direction of displacement.

01:32 So, displacement is in the direction of, along the plane direction, therefore we will resolve mg component in the direction of plane.

01:42 So, mg will have component perpendicular to the plane as mg cost theta.

01:48 So i'm writing here theta, we will later replace this theta with 37 degree.

01:53 And parallel component is mg sine theta.

01:59 All right.

02:00 So what is the value of normal reaction? since there is no motion in the normal direction or perpendicular to the plane direction therefore n will be equals to m g cost theta so what will the value of friction force so friction force will be equals to mu into n since it is a kinetic friction therefore it will be equals to mu into m g cost theta so we have put here the value of n from the above equation now we know the value of friction force and we know the value of umg sine theta so whatever is the work done, whatever is the work done by the friction force on this block and the gravity on this block in the direction of plane, so these two will do the same amount of work that is stored in the spring potential energy.

02:52 Because when this block will slide down by 10 centimeter distance, so this spring will be stretched by the same amount of 10 centimeter...

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