You are testing a new amusement park roller coaster with an...

You are testing a new amusement park roller coaster with an empty car of mass $120 \mathrm{~kg}$. One part of the track is a vertical loop with radius $12.0 \mathrm{~m}$. At the bottom of the loop (point $A$ ) the car has speed $25.0 \mathrm{~m} / \mathrm{s},$ and at the top of the loop (point $B$ ) it has speed $8.0 \mathrm{~m} / \mathrm{s}$. As the car rolls from point $A$ to point $B$, how much work is done by friction?

Key Concepts

Non-Conservative Forces and Work

Non-conservative forces, such as friction, do work that dissipates mechanical energy (often converting it to thermal energy) during a process. This work is path-dependent and results in a net loss of mechanical energy, which must be accounted for when applying energy conservation concepts.

Friction

Friction is a force that opposes the motion of objects in contact and is a prime example of a non-conservative force. In energy analyses, friction does negative work because it removes mechanical energy from the system, converting it into other forms such as heat, and is crucial when determining the total energy balance in mechanics problems.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It is mathematically defined as one half times the mass of the object multiplied by the square of its speed. Changes in kinetic energy indicate acceleration or deceleration, which can be analyzed using the work-energy theorem.

Work-Energy Theorem

The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy. This idea is critical in analyzing situations where forces do work on an object, allowing one to relate the forces acting (through work) with the resulting change in the object’s speed.

Gravitational Potential Energy

Gravitational potential energy is the energy an object has because of its position in a gravitational field. It depends on the object's mass, the gravitational acceleration, and its height above a reference point. Analyzing energy changes when an object moves vertically often involves accounting for changes in gravitational potential energy.

Transcript

00:01 This problem, we have a roller coaster with a massive 120 kilograms going into this loop -to -loop.

00:07 With a velocity initial 25 meters per second, reaches the top at 8 meters per second.

00:14 This has a radius of 12 meters.

00:18 And we're asked to find what is the final, what is the final, excuse me, what is the work due to friction? what is the work due to friction? okay, well, let's set up our energy equation.

00:31 We've got our energy equation.

00:32 We've got the work due to friction plus the initial kinetic energy.

00:39 So the initial kinetic energy, ki, has to be equal to.

00:45 So this is all the inputs.

00:46 And we can assume that friction is actually going to be, we're expecting a negative answer, but because friction is taken away from our system.

00:54 But this is the input is going to be equal to the energy.

00:58 We have the top.

00:59 The energy at the top, the car gains potential energy to gravity.

01:02 So gains ug, and it still has a final kinetic energy, so plus kinetic energy final.

01:12 All right.

01:14 Now, if we rearrange and solve, the work due friction will simply be, we subtract initial from both sides, so it's going to be ug plus kf minus ki.

01:33 Okay, let's work down here...

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