# Question Video: Conservation of Energy of a Body Moving from a Smooth Inclined Plane to a Rough Horizontal Plane Mathematics

The figure shows a body of mass 1/4 kg before it started to slide along the surface. The two surfaces 𝐴𝐵 and 𝐶𝐷 are smooth. However, the horizontal plane 𝐵𝐶 is rough, and its coefficient of kinetic friction is 7/10. If the body started moving from rest, find the distance that the body covered on 𝐵𝐶 until it came to rest. Consider the acceleration due to gravity to be 𝑔 = 9.8 m/s².

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### Video Transcript

The figure shows a body of mass one-quarter kilograms before it started to slide along the surface. The two surfaces 𝐴𝐵 and 𝐶𝐷 are smooth. However, the horizontal plane 𝐵𝐶 is rough, and its coefficient of kinetic friction is seven-tenths. If the body started moving from rest, find the distance that the body covered on 𝐵𝐶 until it came to rest. Consider the acceleration due to gravity to be 𝑔 equals 9.8 meters per second squared.

We can add the labels of mass is equal one-quarter kilogram and 𝜇, the coefficient of kinetic friction, is equal to seven-tenths to the figure. We can apply the principle of conservation of energy, where the total initial energy of the system is equal to the total final energy of the system, to our problem.

Initially, at position 𝐴, the object is at rest, so the only form of energy that the object has is gravitational potential energy. The object comes to rest somewhere along the plane of 𝐵𝐶. Therefore, the final energy does not have any potential energy or kinetic energy, only the work done against friction. We should recall that the gravitational potential energy of an object is equal to the mass of the object times acceleration due to gravity times the height of the object above the ground.

We can substitute in 𝑚𝑔ℎ for the potential energy in our formula. We should also recall that the work done by friction is equal to the force of friction times the distance traveled, which allows us to substitute in force of friction times distance for the work. The problem does not give us the force of friction, but it does give us the coefficient of kinetic friction. So we must use the definition of force of friction, which is the coefficient of kinetic friction times the normal reaction force.

The normal reaction force is the force that a surface puts on an object. When an object is traveling along a horizontal surface as, it does in this problem, without any other vertical forces besides the force of gravity and the normal reaction force acting on it, then we can say the normal reaction force is equal to the force of gravity, where the force of gravity is the mass of the object times the acceleration due to gravity.

Looking at our expanded equation, we can see that there’s an 𝑚 and a 𝑔 on both sides of the equation. Since these are both nonzero numbers, we can cancel them out. Now we can plug in the values from our problem, four for the height and seven-tenths for the coefficient of kinetic friction. To isolate 𝑑, we can multiply both sides of the equation by 10 over seven. This will cancel out the seven-tenths on the right side. The left side of the equation multiplies out to be 40 over seven. The distance the body covered on 𝐵𝐶 until it came to rest is 40 over seven meters.