A sphere rolls down a ramp. If the height of the ramp is h, what is the speed of

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Question: A sphere rolls down a ramp. If the height of the ramp is h, what is the speed of the sphere at the bottom assuming no energy loss?

Options:

Correct Answer: √(2gh)

Solution:

Using conservation of energy, the potential energy at height h converts to kinetic energy at the bottom, giving speed √(2gh).

A sphere rolls down a ramp. If the height of the ramp is h, what is the speed of the sphere at the bottom assuming no energy loss?

A sphere rolls down a ramp. If the height of the ramp is h, what is the speed of the sphere at the bottom assuming no energy loss?

Step 1: Understand that the sphere starts at a height 'h' on the ramp.
Step 2: Recognize that at the top, the sphere has potential energy due to its height.
Step 3: Know that potential energy (PE) can be calculated using the formula PE = mgh, where 'm' is mass, 'g' is the acceleration due to gravity, and 'h' is the height.
Step 4: As the sphere rolls down the ramp, this potential energy converts into kinetic energy (KE) at the bottom.
Step 5: The kinetic energy (KE) of the sphere can be expressed as KE = (1/2)mv^2, where 'v' is the speed of the sphere.
Step 6: Since there is no energy loss, the potential energy at the top equals the kinetic energy at the bottom: mgh = (1/2)mv^2.
Step 7: Cancel the mass 'm' from both sides of the equation, since it appears in both terms.
Step 8: Rearrange the equation to solve for 'v': 2gh = v^2.
Step 9: Take the square root of both sides to find the speed: v = √(2gh).

Conservation of Energy – The principle that energy cannot be created or destroyed, only transformed from one form to another.
Potential Energy – The energy possessed by an object due to its position in a gravitational field, calculated as mgh.
Kinetic Energy – The energy of an object in motion, calculated as (1/2)mv².
Rolling Motion – The motion of a sphere that involves both translational and rotational kinetic energy.