Forces and Kinetic Energy of Rolling A hollow sphere of radius 0.15 m, with rotational inertia I

Forces and Kinetic Energy of Rolling A hollow sphere of...

Forces and Kinetic Energy of Rolling A hollow sphere of radius $0.15 \mathrm{m},$ with rotational inertia $I=0.040 \mathrm{kg} \cdot \mathrm{m}^{2}$ about a line through its center of mass, rolls without slipping up a surface inclined at $30^{\circ}$ to the horizontal. At a certain initial position, the sphere's total kinetic energy is 20 J. (a) How much of this initial kinetic energy is rotational? (b) What is the speed of the center of mass of the sphere at the initial position? When the sphere has moved 1.0 $\mathrm{m}$ up the incline from its initial position, what are (c) its total kinetic energy and (d) the speed of its center of mass?

Forces and Kinetic Energy of Rolling
A hollow sphere of radius $0.15 \mathrm{m},$ with rotational inertia $I=0.040 \mathrm{kg} \cdot \mathrm{m}^{2}$ about a line through its center of mass, rolls without slipping up a surface inclined at $30^{\circ}$ to the horizontal. At a certain initial position, the sphere's total kinetic energy is 20 J. (a) How much of this initial kinetic energy is rotational? (b) What is the speed of the center of mass of the sphere at the initial position? When the sphere has moved 1.0 $\mathrm{m}$ up the incline from its initial position, what are (c) its total kinetic energy and (d) the speed of its center of mass?

Key Concepts

Translational Kinetic Energy

Translational kinetic energy is the energy associated with the motion of the center of mass of a body. It is given by (1/2)mv² and represents the energy due to the linear motion of the object. This concept is key when analyzing systems where parts of the energy are due to the movement of the whole body along a path, such as a sphere rolling on an incline.

Rotational Kinetic Energy

Rotational kinetic energy is the energy due to the rotation of a body about its axis. It is calculated using (1/2)I?², where I is the moment of inertia and ? is the angular velocity. This energy component is essential in dynamics problems involving rolling objects, as part of the total kinetic energy is stored in the rotation of the object.

Rolling Without Slipping

Rolling without slipping is a constraint that links the translational speed of the center of mass and the angular speed of the rolling body. Mathematically, it is expressed as v = ?R, where v is the linear speed, ? is the angular speed, and R is the radius of the rolling object. This condition ensures that the point of contact between the object and the surface is momentarily at rest, which is critical for analysing the energy distribution in rolling motion.

Moment of Inertia

The moment of inertia is a measure of how an object's mass is distributed relative to its axis of rotation. It influences how resistant an object is to changes in its rotational motion. In problems involving rolling, the moment of inertia determines how much of the total kinetic energy is partitioned into rotational motion compared with translational motion.

Conservation of Energy in Rolling Motion

The principle of conservation of energy states that the total energy in an isolated system remains constant, provided no energy is lost to friction or other non-conservative forces. In the context of a rolling object on an incline, this means that the sum of translational and rotational kinetic energies, along with gravitational potential energy, remains constant as the object moves, thus linking changes in kinetic energy to changes in height or potential energy.

Transcript

00:01 Problem 11 .10.

00:04 So we have a hollow sphere rolling without slipping up an incline at 30 degrees its size and moment of inertia are given here and it has a kinetic energy at some initial point of 20 joules so first we want to answer how much of the kinetic energy is rotational so a rotational over the center of mass motion related kinetic energy plus k rotational so for a hollow sphere the moment of inertia is two -thirds m r squared and it may be handy for us to know that the mass must be 2 .7 kilograms so one half i.

01:33 Omega squared based on this is one third mr squared omega squared.

01:54 Now we know that v is omega times r.

01:58 So our kinetic energy from the center of mass motion here can be written as 1 half m v squared, but v squared is also r squared omega squared because it's not slipping, plus just the exact same thing here is on top, 1 3rd m, r squared, omega squared.

02:27 And so canceling out all the mr squared, omega squared, we're left with the ratio of 0 .4.

02:39 So 40 % of its kinetic energy is rotational.

02:51 So we now want to know what's the initial speed of the center of mass.

03:02 So we know that the center of mass kinetic energy is 40 % of 20, which is 8.

03:10 This is 8 joules.

03:23 This is the rotational kinetic energy.

03:31 You can use either one, but you need to get the right number for the right thing.

03:39 So this is 1 half m r squared, i'm sorry, no, not 1 half.

03:50 If you find physics confusing, don't worry.

03:54 Everyone does...