Aim for FivePhysics 1

Translation between representations (TBR)

Racing a sphere and a hoop down a ramp

  • Units 5 and 6
  • 12 points
  • About 28 minutes

You can use a calculator on this question, just like on exam day.

A multipart problem about one situation shown in several ways. You draw a diagram, derive equations, sketch or draw graphs, and then explain whether your answers agree with each other or use them to predict what happens when the situation changes. On the exam: Question 2 of 4. Section II has 4 questions in 95 minutes (50% of the exam score), one of each type in this order; calculator allowed. The CED suggests 25–30 minutes.

The question

A solid sphere (rotational inertia (2/5)MR² about its center) and a thin hoop (rotational inertia MR² about its center) have the same mass M and the same radius R. Both are released from rest at the same time from the top of the same incline, which makes an angle θ with the horizontal and has vertical height h. Both roll without slipping. Air resistance is negligible.

Suggested time: 28 minutes

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Part (a)

2 points

Describe a force diagram for the sphere as it rolls down the incline. For each force, give the direction and the point on the sphere where it is exerted.

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Part (b)

3 points

For an object with rotational inertia βMR² (β is a number), derive an expression for the speed at the bottom of the incline in terms of β, g, and h. Then write the result for the sphere and for the hoop.

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Part (c)

2 points

Describe the energy bar charts for the sphere–Earth system and the hoop–Earth system at the bottom of the incline, showing translational kinetic energy and rotational kinetic energy as fractions of Mgh.

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Part (d)

3 points

On a single set of axes, describe the graphs of speed versus time for the sphere and the hoop, from release until each reaches the bottom.

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Part (e)

2 points

Both objects have the same mass and lose the same gravitational potential energy. Explain why the hoop reaches the bottom later, using your bar charts from part (c).

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