A hollow cylinder (hoop) is rolling on a horizontal surface at...

Question

A hollow cylinder (hoop) is rolling on a horizontal surface at speed v = 3.0 m/s when it reaches an 18° incline. How far up the incline will it go?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice from together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A solid sphere is rolling on a horizontal surface with an initial speed of V equals 4.5 m per second when it encounters a ramp inclined at 25 degrees. What distance will it ascend? The incline before coming to a stop? Assume no slippage occurs during the motion. So that's our end goals. We're trying to figure out when this ball is rolling on this horizontal surface and it encounters an incline, how far up the incline it will go before coming to a stop. So I'll go up the incline and we're trying to figure out where it will stop on the incline before it starts rolling back down the ramp. OK. So we're also given some multiple choice answers. They're all in the same units of meters. So let's read them off to see what our final answer might be. A is 1.9 B is 2.0 C is 3.4 D is 4.8. Awesome. So first off, let us create a diagram to help us better visualize this problem. So I've gone ahead and drawn a diagram to help us better visualize what is happening in this particular problem. So as we can see, we have a dotted line which represents the horizontal plane and then we have a solid diagonal black line to represent the incline that's 25 degrees. So to make things simple, I drew a sphere since it's a ball as a sphere, say it shape or I'm calling it a ball but a solid spear. So we have our solid sphere in blue when it's on the horizontal plane. And to avoid confusion, I drew the sphere or it comes to some stop on the incline in red. So it's gonna stop somewhere on this incline before it decides to roll back down to the horizontal plane, the horizontal surface where it started out. So the velocity is 4.5 m per second to the right. So it's gonna go on the horizontal surface, go, go, go and then it's gonna reach the ramp, roll up the ramp and it's gonna come to a stop for a brief second before it rolls back down. And the incline is some height. H OK. Now that we have a visualization of what's going on here, we now need to assume that there is no dissipative dissipative forces, so no dissipative forces doing work. Therefore, we can recall and use the conservation of mechanical energy principle to help us solve for the final height H that the sphere reaches on this incline, which is the height H of the spirit, which is what we need to use in order to solve for D the distance or final answer. So we have to find out what the value of H is before we can solve for the value of D. Which is our final answer. We also need to consider that the bottom of the incline is to be the zero level of the gravitational potential energy. Also, since the sphere is rolling without sliding, we can therefore write that the angular frequency is so we can write the equation for the angular frequency as omega is equal to V divided by R where V is the linear frequency and R is the radius of the sphere. Let us note that the sphere has both translational and rotational kinetic energy at the bottom. Therefore, we can recall and use the equation for the inertia of a sphere which the inertia of a sphere which or rather the moment of inertia for a sphere is denoted as I and the moon inertia of a sphere is equal to 2/5. So we need to take 2/5 and we need to multiply it by the mass of the sphere multiplied by the radius of the sphere squared. Awesome. So now we need to recall and apply the conditions of the conservation of energy to the top and bottom of the incline. Thus, we can write that the energy at the top, which I'm gonna denote it as E subscript T is equal to E subscript B. So once again to reiterate here, the energy at the top, so the energy at the top of the incline and the bottom of the incline are equal to each other due to the conservation of energy. So, et is equal to ev so the energy at the top is equal to the energy at the bottom. So therefore, we can also write that and I'll do a little arrow here. We can also take it a step further. And we could write that M multiplied by G multiplied by H is equal to one half multiplied by M multiplied by V squared plus one half multiplied by I multiplied by omega square, which we can simplify and write that M multiplied by G multiplied by H is equal to one half multiplied by M multiplied by V squared plus one half multiplied by two, divided by five or 2/5 multiplied by M multiplied by R squared multiplied by V divided by R. And this value is all squared. So V divided by R is all squared. Awesome. So what we did here is we plugged in all of our known variables. OK. Awesome. Moving right along here. So we need to note that we are trying to rearrange our conservation of energy equation to isolate and solve for H all by itself. So when we start to do that, we will get that G multiplied by H is equal to one half multiplied by V squared plus 1/5 multiplied by V squared. And we can simplify this and we can say that G multiplied by H is equal to 0.7 multiplied by V squared. Finally getting H all by itself, we will get that H is equal to 0.7 multiplied by V squared, all divided by G. Now we can plug in all of our known variables and solve for the numerical value of H. So we need to take 0.7 and we need to multiply it by our value of V which we know that the speed is 4.5 in its meters per second. And don't forget that our velocity is squared multiplied by the acceleration due to gravity, which numerically that is 9.81 m per second squared as one should recall. And this is all equal to, let me plug it into our calculator, 1.4 4 5 m when we round to three decimal places. So now at this point, we can solve for D, the distance, which is our final answer. And this is the distance along the plane which we can find this value by recalling and using trigonometry which states that D our distance is equal to the height H divided by sine of 25 degrees because as we know, our incline is 25 degrees. So now if we plug in our known values, we know that the height is 1.445 m, which is the value we just found together a moment ago. And we need to divide this by sine of 25 degrees. So when we plug that into our calculator, we will get 3.4 m and this is when we round to one decimal place and that's it we've solved for this problem. Hooray. So looking at our multiple choice answers, the correct answer has to be the letter C 3.4 m. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.

A hollow cylinder (hoop) is rolling on a horizontal surface at speed v = 3.0 m/s when it reaches an 18° incline. How far up the incline will it go?

Key Concepts

Conservation of Energy

Kinetic Energy

Inclined Plane Dynamics

Watch next