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A ball rolls down a ramp. If it starts from rest and rolls without slipping, wha

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Question: A ball rolls down a ramp. If it starts from rest and rolls without slipping, what is the relationship between its linear speed and angular speed at the bottom?

Options:

  1. v = Rω
  2. v = 2Rω
  3. v = R/2ω
  4. v = 3Rω

Correct Answer: v = Rω

Solution: 

The relationship is given by v = Rω, where v is the linear speed, R is the radius, and ω is the angular speed.

A ball rolls down a ramp. If it starts from rest and rolls without slipping, wha

Practice Questions

Q1
A ball rolls down a ramp. If it starts from rest and rolls without slipping, what is the relationship between its linear speed and angular speed at the bottom?
  1. v = Rω
  2. v = 2Rω
  3. v = R/2ω
  4. v = 3Rω

Questions & Step-by-Step Solutions

A ball rolls down a ramp. If it starts from rest and rolls without slipping, what is the relationship between its linear speed and angular speed at the bottom?
  • Step 1: Understand that the ball is rolling down a ramp without slipping. This means that the ball's rotation is directly related to how fast it moves forward.
  • Step 2: Identify the terms: 'v' is the linear speed (how fast the ball moves in a straight line), 'R' is the radius of the ball (how big the ball is), and 'ω' is the angular speed (how fast the ball is spinning).
  • Step 3: Realize that when the ball rolls without slipping, there is a specific relationship between these speeds. This relationship is expressed by the formula v = Rω.
  • Step 4: In this formula, if you know the radius of the ball and its angular speed, you can calculate its linear speed. Conversely, if you know the linear speed and the radius, you can find the angular speed.
  • Rolling Motion – The relationship between linear speed and angular speed for an object rolling without slipping.
  • Moment of Inertia – Understanding how the distribution of mass affects the motion of rolling objects.
  • Conservation of Energy – The conversion of potential energy to kinetic energy as the ball rolls down the ramp.
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