A 20.0-kg rock is sliding on a rough, horizontal surface at 8.00 m/s and eventually stops due to

A 20.0-kg rock is sliding on a rough, horizontal surface at...

Key Concepts

Power

Power is defined as the rate at which work is done or energy is transferred over time. It is calculated by dividing the work done by the time interval over which the work is performed. In scenarios of energy dissipation, such as an object coming to rest due to friction, average power measures how quickly kinetic energy is converted into other forms, primarily heat.

Work-Energy Principle

The work-energy principle states that the net work done on an object is equal to its change in kinetic energy. In systems where friction is the dominant force opposing motion, the work done by friction (which is negative) corresponds to the loss of the object’s kinetic energy as it comes to a stop.

Kinetic Energy

Kinetic energy is the energy associated with the motion of an object, given by the formula KE = 1/2 m v^2. It quantifies the work required to accelerate an object from rest to its current velocity, and its change indicates the mechanical energy transformation taking place, such as conversion to heat when friction is involved.

Friction

Friction is the resistive force that acts opposite to the direction of motion when two surfaces slide past each other. In problems of motion on rough surfaces, friction—quantified by the coefficient of kinetic friction—plays a key role in slowing down and eventually stopping the moving object by converting kinetic energy into thermal energy.

Transcript

00:01 Welcome to our new video where we want to compute the power involved in stopping a rock.

00:09 So let's say this rock is moving eastward at a velocity of 8 .0 meters per second.

00:17 So it has motion.

00:18 That means it also has kinetic energy.

00:21 The mass of the rock happens to be 20 .0 kilograms.

00:26 Eventually, it's going to move and come to a stop.

00:30 So the final velocity when it stops is going to be zero.

00:35 The acceleration due to gravity is 9 .8 meters per second squared.

00:40 The reason why the rock stops is cause of this rough surface.

00:45 So there is kinetic friction opposing it with the coefficient of kinetic friction being 0 .2 .0.

00:55 The other thing that happens is that we can always compute the friction of force.

01:00 So fk equals to mu k times n.

01:05 Remember, n is the normal force on the rock as a result to the weight, mg.

01:12 So n and mg are equal.

01:15 So the frictional force is mu k, m g.

01:20 We need to get the time it takes for the rock to stop from its initial position to the final position.

01:27 The reason for that is power, the average power, is a, energy over time.

01:35 So energy over time, but in this case we're dealing with kinetic energy over time.

01:40 Kinetic energy is the energy that objects have as a result of their motion.

01:46 So then to get the power, we're just going to do one half mv squared over the time.

01:52 We do have the velocity, we do have the mass, but we don't have the time it takes for the object to stop.

01:58 So using the equations of motion, we know that the final velocity, which happens to be zero at this point, zero meters per second, is given by initial velocity plus the acceleration times the time.

02:15 So our goal is to get the time it takes for the object to stop, but then we don't have the acceleration or the deceleration of the object.

02:29 So we're missing two variables, the acceleration or deservation and the time.

02:37 If you look at the frictional force, we can use newton's second law, which says the net force is m .a.

02:45 Or the mass time's acceleration.

02:46 So this simply means that the force on an object produces a specific acceleration.

02:55 In this case, it's producing a desereration...

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