A sphere rolls on a flat surface with a speed v. What is the kinetic energy of t

₹0.0

Question: A sphere rolls on a flat surface with a speed v. What is the kinetic energy of the sphere?

Options:

Correct Answer: (1/2)mv^2 + (2/5)mv^2

Solution:

The total kinetic energy of a rolling sphere is the sum of translational and rotational kinetic energy, which is (1/2)mv^2 + (2/5)mv^2.

A sphere rolls on a flat surface with a speed v. What is the kinetic energy of the sphere?

A sphere rolls on a flat surface with a speed v. What is the kinetic energy of the sphere?

Step 1: Understand that a sphere rolling on a flat surface has two types of motion: it moves forward (translational motion) and it spins (rotational motion).
Step 2: The formula for translational kinetic energy (the energy due to forward motion) is (1/2)mv^2, where m is the mass of the sphere and v is its speed.
Step 3: The formula for rotational kinetic energy (the energy due to spinning) for a solid sphere is (2/5)mv^2.
Step 4: To find the total kinetic energy of the rolling sphere, add the translational kinetic energy and the rotational kinetic energy together.
Step 5: The total kinetic energy is (1/2)mv^2 + (2/5)mv^2.

Translational Kinetic Energy – The energy due to the motion of the center of mass of the sphere, calculated as (1/2)mv^2.
Rotational Kinetic Energy – The energy due to the rotation of the sphere about its center of mass, calculated as (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.
Moment of Inertia – For a solid sphere, the moment of inertia I is (2/5)mr^2, which is used to calculate the rotational kinetic energy.
Rolling Motion – Understanding the relationship between translational and rotational motion in rolling objects.