A bicycle racer is going downhill at 11.0 ms when, to his horror, one of his 2.25-kg wheels comes off as he is 75.0 m above the foot of the hill. We can model the wheel as a thin-walled cylinder 85.0 cm in diameter and ignore the small mass of the spokes. (b) How much total kinetic energy does the wheel have when it reaches the bottom of the hill?

Question

A bicycle racer is going downhill at 11.0 m>s when, to his horror, one of his 2.25-kg wheels comes off as he is 75.0 m above the foot of the hill. We can model the wheel as a thin-walled cylinder 85.0 cm in diameter and ignore the small mass of the spokes. (b) How much total kinetic energy does the wheel have when it reaches the bottom of the hill?

Accepted Answer

Welcome back. Everyone in this problem. A roller blader moves downhill on an inclined plane with an initial velocity equal to 9 m per second. Suddenly one wheel weighing around 0.12 kg got detached while he was still 65 m above ground level. This wheel can be modeled as a thin cylinder with a diameter equal to 60 centimeters, ignoring any other attachments on it. What will be the total kinetic energy of this wheel once it hits ground level? A says it's 55 Joules B 86 Joules C 96 Joules and D 150 Joules. Now, if we are going to figure out the total kinetic energy of our wheel, there are a few things that we need to establish. So first, let's go ahead and make note of all the information we have because it's quite a lot. So first of all, we know that initially our velocity is 9 m per second. The masa four wheel is 0.12 kg. The initial height ok is 65 m. And because our, we know our our roller blader is going downhill and eventually it it the wheel hits ground level. So the final height at ground level is going to be 0 m. We also know that the diameter of this wheel is 60 centimeters, which tells us then that the radius is going to be a half of that or 30 centimeters. Or we can write that as 0.3 m. And from this, we want to figure out the total kinetic energy of this wheel once it hits ground level. In other words, we're solving for our final kinetic energy KF. Now how is our kinetic energy related to all the information we have here? Well, first, let's make a note of what we know about kinetic energy. Recall that our total kinetic energy for our wheel. OK? Is going to be a combination of the translational the kinetic energy due to translational motion. Sorry. And the kinetic energy due to rotational motion for our translational motion, the kinetic energy is going to be half MV squared. And for our rotational motion, it's going to be half I omega squared where I is the moment of inertia and omega is our angular velocity. Now, here we're talking about our initial velocity. OK. And for the wheels, for the wheels to roll without slipping or for the wheel that rose without slipping. OK? Because it didn't say that it slips here then OK. Its initial velocity is equal to the angular velocity multiplied by the radius R. But in this case, that tells us then that our angular velocity is going to be equal to the initial velocity V divided by R. On the other hand, we also know that since the wheel is assumed to be a thin cylinder, then its moment of inertia eye is going to be equal to Mr squared. So with both of these, we can substitute that into our formula for our total kinetic energy, which means that K is going to be equal to a half of MV I squared plus a half of the moment of inertia M squared. OK. Multiplied by our angular velocity V divided by R or squared. Now, we can do some simplification here because if we expand or square for our bracket, OK, then it's gonna be a half of MV I squared plus a half of M multiplied by R squared, multiplied by V I squared divided by R squared and R squared cancels R squared to give us a total kinetic energy of a half MV squared plus a half MV squared. I know when we add both of these, our total kinetic energy then is going to be MV squared. Know that we have an expression for total kinetic energy, then we can apply the conservation of energy and by the law of conservation of energy, it tells us that our initial energy is going to be equal to our final energies. Now, in this case for our rolling wheel, we know that the energies involved here are the kinetic energy and the gravitational potential energy. So by the conservation of energy, then the initial kinetic energy and the initial gravitational potential energy is equal to the final kinetic and gravitational potential energies. OK. No, in this case, we know that our final gravitational potential energy since it hits a ground level is going to be zero. OK. So that means the final kinetic energy equals the initial kinetic energy plus the initial gravitational potential energy. Now, what do we know about all of these? Well, we know that let me put that in red here, that initial kinetic energy is going to be equal to initial kinetic energy is going to be MV I squared. And that initial gravitational potential energy equals MGH I OK. Where M is the mass G is the acceleration due to gravity? And hh I is our initial height. Now, since we have all of this information, that means that we should be able to solve our final kinetic energy. Because now here if we substitute, well, first, if we factored M that's going to be M multiplied by V squared plus GH OK. And that's equal to our mass of 0.12 kg. That's the mass of the wheel. OK. Multiplied by uh our initial speed. Our initial speed of 9 m per second squared. OK. Sorry 9 m per second. My apologies. All squared, all of that is squared plus our acceleration due to gravity which we can take as 9.8 m per second squared, multiplied by our initial height of 65 m. Thus, this is how we'll be able to find the final kinetic energy. Now, when we go ahead and substitute these values into our calculator, ok, then we should get the final kinetic energy to be equal to 86.16 joules or rounded to two significant figures, approximately 86 joules. Therefore, that's our total kinetic energy at the bottom. And when we go back to our answer choices that tells us that B must be the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Verified Solution

Key Concepts

Kinetic Energy

Potential Energy

Conservation of Energy