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AP Physics C: Mechanics · Question 9
AP Physics C: Mechanics · Original GioPhysics question with a detailed, mark-by-mark answer guide.
- Demand
- demanding
- Marks
- 11
- Topics
- 2
- Answer
- Complete
A student wants to determine experimentally the moment of inertia I of a wheel that is free to rotate about a fixed horizontal axle. A light string is wound around a hub of known radius r on the axle, and a hanging mass m is attached to the free end. When released, the mass falls and the wheel rotates. Friction in the axle is not negligible.
- (a)
Describe Describe a procedure for measuring the linear acceleration of the falling mass, and state the quantity the student should deliberately vary between trials.
4 marks - (b)
Derive Ignoring friction for this part, derive a relationship between the acceleration a of the hanging mass and the moment of inertia I, and describe how the student should plot the data to obtain a straight line.
4 marks - (c)
Explain Explain how the presence of friction in the axle would affect the value of I obtained from the slope, and describe a modification to the analysis that would account for it.
3 marks
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(a)
Describe Describe a procedure for measuring the linear acceleration of the falling mass, and state the quantity the student should deliberately vary between trials.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
1 point: release the mass from rest from a measured height h and time the fall with a stopwatch or photogate
- 2
1 point: obtain the acceleration from h = ½at², repeating each trial and averaging the time
- 3
1 point: vary the hanging mass m between trials, keeping the hub radius and the wheel unchanged
- 4
1 point: use a photogate or video rather than a hand-operated stopwatch where possible, since the fall times are short and reaction time is a significant fraction of them
(b)
Derive Ignoring friction for this part, derive a relationship between the acceleration a of the hanging mass and the moment of inertia I, and describe how the student should plot the data to obtain a straight line.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
1 point: for the hanging mass, mg − T = ma; for the wheel, Tr = Iα with α = a/r
- 2
1 point: eliminating T gives mg = ma + Ia/r², so a = mg/(m + I/r²)
- 3
1 point: rearranges to a linear form — for example 1/a = (I/r²)(1/(mg)) + 1/g
- 4
1 point: plot 1/a against 1/m; the slope is I/(gr²) and the vertical intercept is 1/g
(c)
Explain Explain how the presence of friction in the axle would affect the value of I obtained from the slope, and describe a modification to the analysis that would account for it.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
1 point: a frictional torque opposes the rotation, so every measured acceleration is smaller than the frictionless model predicts
- 2
1 point: the analysis attributes that reduction entirely to rotational inertia, so the value of I obtained is too large
- 3
1 point: include a constant frictional torque τ_f in the wheel's equation, Tr − τ_f = Iα, which adds a constant term to the linearised relationship and can be found from the intercept — or measure the torque needed to keep the wheel turning at constant speed and subtract it
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