Question

A solid cylinder, a hollow cylinder, a solid sphere, and a hollow sphere are rolling without slipping. All four objects have the same mass and radius and are traveling with the same linear speed. Which one of the following statements is true? a) The solid cylinder has the highest kinetic energy. b) The hollow cylinder has the highest kinetic energy. c) The solid sphere has the highest kinetic energy. d) The hollow sphere has the highest kinetic energy. e) All four objects have the same kinetic energy.

Short answer

Answer: The hollow cylinder has the highest kinetic energy.

Step-by-step solution

Identify the moment of inertia for each object

For each object, we need to find the moment of inertia to compare their kinetic energy. The moment of inertias for these objects are given by the following formulas (assuming all objects are rotating around their central axes): - Solid cylinder (SC): I_SC = (1/2) * M * R^2 - Hollow cylinder (HC): I_HC = M * R^2 - Solid sphere (SS): I_SS = (2/5) * M * R^2 - Hollow sphere (HS): I_HS = (2/3) * M * R^2

Calculate the rotational kinetic energy for each object

The rotational kinetic energy for each object is given by the formula K_rot = (1/2) * I * ω^2, where I is the moment of inertia, and ω is the angular velocity. Since all objects have the same linear speed (v), we can use the relation ω = v / R and rewrite the formula as follows: K_rot = (1/2) * I * (v^2 / R^2) Now, substitute the moments of inertia (from Step 1) for each object: - Solid cylinder (SC): K_rot_SC = (1/2) * (1/2) * M * R^2 * (v^2 / R^2) = (1/4) * M * v^2 - Hollow cylinder (HC): K_rot_HC = (1/2) * M * R^2 * (v^2 / R^2) = (1/2) * M * v^2 - Solid sphere (SS): K_rot_SS = (1/2) * (2/5) * M * R^2 * (v^2 / R^2) = (1/5) * M * v^2 - Hollow sphere (HS): K_rot_HS = (1/2) * (2/3) * M * R^2 * (v^2 / R^2) = (1/3) * M * v^2

Compare the rotational kinetic energies to find the object with the highest kinetic energy

We have found the rotational kinetic energy for each object. Let's compare the values to see which has the highest kinetic energy: - K_rot_SC = (1/4) * M * v^2 - K_rot_HC = (1/2) * M * v^2 - K_rot_SS = (1/5) * M * v^2 - K_rot_HS = (1/3) * M * v^2 Note that M and v^2 are constant for all objects. (1/2) is greater than (1/4), (1/5), and (1/3). Therefore, the object with the highest kinetic energy is the hollow cylinder. Answer: b) The hollow cylinder has the highest kinetic energy.

Key concepts

Moment of Inertia

The moment of inertia, often represented as 'I', is a quantitative measure of an object's resistance to changes in its rotational motion. Imagine trying to spin a bicycle wheel by pushing on its pedals—the amount of effort you need to exert is closely related to the wheel's moment of inertia. The larger the moment of inertia, the harder it is to change the rotational speed of the object.

In the case of objects rolling without slipping—like our cylinders and spheres—the moment of inertia depends on the mass distribution relative to the axis of rotation. Here's an easy way to picture this: a hollow cylinder has its mass concentrated at a distance from the center, so it has a greater moment of inertia than a solid cylinder of the same mass and size, which has some of its mass closer to the axis.

For our exercise, the various moments of inertia were calculated using standard formulas. When we compare objects of the same mass and radius, the differences in the moment of inertia directly impact their rotational kinetic energy. A higher moment of inertia would usually mean less kinetic energy for the same amount of rotational effort (angular velocity), but here we must also consider the precise relationship between inertia and the objects’ kinetic energy calculations.

Rolling Without Slipping

When an object is 'rolling without slipping', it means that the object rolls in such a way that its point of contact with the ground does not slide. One of the easiest ways to picture this is to think about a tire moving on a road: when the tire is not skidding, it is rolling without slipping. This concept is crucial for our problem because it links linear motion (like the tire moving forward on the road) with rotational motion (the spinning of the tire).

In physics, when an object rolls without slipping, a special relationship exists between the linear velocity of the object's center of mass (let's call it 'v') and its angular velocity (denoted by the Greek letter 'ω'). This relationship is expressed as 'ω = v / R', where 'R' is the radius of the rolling object. This connection is vital to solve our problem since it allows us to equate the linear speed of the rolling objects to their angular speed. Thus, when comparing kinetic energy, we can directly apply this relation to find the rotational kinetic energy using the angular velocity, captured in the formula 'K_rot = (1/2) * I * (v^2 / R^2)'.

Angular Velocity

Angular velocity represents the rate at which an object rotates about its axis. Assuming you've watched figure skating, you'll notice how skaters spin faster when they pull their arms close to their body—they're reducing their moment of inertia which, according to the conservation of angular momentum, increases their angular velocity.

In our problem, angular velocity is directly proportional to the linear speed since the objects are rolling without slipping. This means that if two objects have the same linear speed but different radii, the one with the smaller radius will have a higher angular velocity. It's important to note that while angular velocity is critical in calculating rotational kinetic energy 'K_rot', the linear speed 'v' is constant for all the objects in our exercise, simplifying our calculations.

The formula for rotational kinetic energy depends on the square of the angular velocity, meaning that differences in angular velocity can lead to significant differences in kinetic energy. This is why objects with different moments of inertia still can have vastly differing kinetic energies despite having the same linear velocity, as angular velocity acts as a great equalizer in our formula.