A solid uniform sphere and a thin-walled, hollow sphere have the same mass M and radius R . If t

A solid uniform sphere and a thin-walled, hollow sphere have...

Key Concepts

Rolling Without Slipping

This concept refers to the constraint where an object rolls in such a way that there is no relative motion between the point of contact and the surface. In rolling motion, the linear speed of the center?of-mass is directly related to the angular speed about the center through the radius of the object. This relationship is crucial in analyzing the dynamics of rolling objects because it links rotational and translational motions, allowing us to account for energy partitioning correctly.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to rotational acceleration about a given axis and depends on how its mass is distributed relative to the axis of rotation. In comparing different objects, such as a solid sphere versus a thin-walled hollow sphere, differences in the moment of inertia lead to differences in the amount of energy stored as rotational kinetic energy when rolling. A larger moment of inertia for the same mass and radius means that more energy is ‘invested’ in rotation, which affects how much energy is available to convert to gravitational potential energy when moving up an incline.

Translational and Rotational Kinetic Energy

In rolling objects, the total kinetic energy is the sum of translational kinetic energy (associated with the motion of the center of mass) and rotational kinetic energy (due to spinning about the center). For an object rolling without slipping, the rotational kinetic energy is determined by the moment of inertia and the angular speed. This energy partitioning is critical because, when an object climbs a ramp, the conversion of kinetic energy to gravitational potential energy depends on the total energy the object initially has.

Energy Conservation

Energy conservation is the principle that the total mechanical energy of a system remains constant if only conservative forces (like gravity) are doing work. In the context of a rolling object on an incline, the initial total kinetic energy, which includes contributions from both translation and rotation, is converted into gravitational potential energy as the object ascends. Differences in the distribution of kinetic energy between objects with different moments of inertia determine the maximum height reached by each.

Transcript

00:01 In this problem, we're comparing two balls rolling up a hill of inclined beta.

00:05 They have the same mass and radius.

00:08 The only difference is that one is a solid sphere and one is a hollow sphere.

00:13 And based on that, and our knowledge of forces and torques, we are supposed to find the height each one achieves rolling up the hill.

00:22 So to break this down, the first thing i'm going to do is find the coefficient of friction in each case, because we aren't ignoring friction in this problem.

00:33 And in order to do that, i'm going to make just a catch -all formula to find the coefficient of friction based on these parameters.

00:43 And to start that, i've broken down these moment of inertia equations into the moment of inertia of a ball.

00:51 It's going to be equal to a constant times the mass, times the radius squared.

00:58 So i'll be using that in this formula.

01:00 We already know to find the coefficient friction we need to compare forces and torques so we know that the the total force acting on the sphere is equal to the downhill force plus the force of friction so this will be m a and this will be m g sine beta and plus f and f and f we know is what's enacting the only torque on this.

01:37 So we can use our torque equation.

01:41 F .r.

01:43 And that will be equal to i alpha or ia over r.

01:50 And we can just divide by r.

01:52 We get f equals ia over r squared.

01:58 And we can plug in our formula here.

02:02 So this will be kmr squared a over r squared.

02:11 These will cancel, and we will be left with sine beta plus kma.

02:25 So now all we need to do is subtract that, and we get ma times 1 minus k is equal to mg sine beta.

02:40 And then we can cancel up this m here, divide by 1 over k, and we will get a is equal to g sine, beta over 1 minus k okay so now we substitute that that back into this equation so this will be mg sine beta times 1 over 1 minus k is equal to m g sine beta sign beta plus f now we're going to substitute in what we know for f which is the coefficient of friction times the normal force and we know that the normal force is going to be m .g.

03:30 Cosine beta.

03:33 So we pull that out on the left.

03:37 M .g.

03:39 Cosine beta is equal to 1 over 1 minus k.

03:48 M .g.

03:51 Sine beta minus m g sine beta, which will be equal to mg sine beta, times 1 over 1 minus k minus 1.

04:12 And this, we will change this 1 to 1 minus k over 1 minus k.

04:20 And then subtract this out to get our coefficient here, k over 1 minus k.

04:32 And then we'll just divide by this side.

04:34 And we will get the coefficient of friction is equal to m g, sine beta over m g cosine beta cancel these out and add our coefficient k over one minus k okay so i am going to just put that off to the side and clear my workspace so we can start on the calculation of the height that this ball will achieve now that we that coefficient of friction, this will be much simpler.

05:30 So we know that the energy that each ball has, it's going to be equal to 1 .5 mv squared, the kinetic energy, plus 1 .5 i.

05:44 Omega squared, the rotational kinetic energy.

05:48 And that will, of course, be equal to mgh.

05:52 If we attain this height, then all of the energy will be going here.

05:57 But there's one more place that energy is going, and that is into friction.

06:02 And of course, that'll just be the work friction does, which will be f times the distance.

06:09 So let's break down f again.

06:12 We know mgh plus f is going to be equal to the normal force times the coefficient of friction.

06:23 I'll bring up the coefficient of friction in front times mg cosine theta.

06:31 Not theta, beta, times the distance...

Please give Ace some feedback