IB · A · structured
Space, time and motion · Question 10
Space, time and motion · Original GioPhysics question with a detailed, mark-by-mark answer guide.
- Demand
- discriminating
- Marks
- 14
- Topics
- 2
- Answer
- Complete
Part 1. A uniform solid cylinder of mass 2.0 kg and radius 0.15 m rolls without slipping down a slope, starting from rest at a height of 1.2 m above the bottom. The moment of inertia of a uniform solid cylinder about its axis is ½MR². Part 2. A spacecraft passes Earth at a constant speed of 0.80c. The spacecraft has a proper length of 90 m.
- (a)
Show (that) Show that the translational speed of the cylinder at the bottom of the slope is about 4.0 m s⁻¹.
4 marks - (b)
Determine Determine the fraction of the cylinder's total kinetic energy at the bottom that is rotational.
3 marks - (c)
Calculate Calculate the length of the spacecraft as measured by an observer on Earth.
3 marks - (d)
Explain An observer on the spacecraft claims that it is the Earth that is 0.80c and that Earth's distances are contracted, not the spacecraft's. Explain why both observers are correct, and outline what would have to change for one of them to be wrong.
4 marks
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(a)
Show (that) Show that the translational speed of the cylinder at the bottom of the slope is about 4.0 m s⁻¹.
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
Mgh = ½Mv² + ½Iω², with I = ½MR²
- 2
applies the rolling condition ω = v/R, so ½Iω² = ¼Mv²
- 3
Mgh = ¾Mv², so v = √(4gh/3)
- 4
v = √(4 × 9.81 × 1.2 / 3) = 3.96 ≈ 4.0 m s⁻¹
(b)
Determine Determine the fraction of the cylinder's total kinetic energy at the bottom that is rotational.
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
rotational KE = ¼Mv², translational KE = ½Mv²
- 2
Adding the translational and rotational terms gives total kinetic energy = ¾Mv².
- 3
fraction rotational = (¼)/(¾) = 1/3
(c)
Calculate Calculate the length of the spacecraft as measured by an observer on Earth.
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
γ = 1/√(1 − 0.80²) = 1/0.60 = 1.67
- 2
L = L₀/γ = 90/1.67
- 3
L = 54 m
(d)
Explain An observer on the spacecraft claims that it is the Earth that is 0.80c and that Earth's distances are contracted, not the spacecraft's. Explain why both observers are correct, and outline what would have to change for one of them to be wrong.
State the outcome first, then link cause to effect with the relevant physical principle. Each link in the reasoning should be explicit enough to earn its own marking point.
- 1
both frames are inertial, and the principle of relativity states that the laws of physics are the same in all inertial frames
- 2
there is no experiment either observer can do to establish that they are the one "really" moving, so neither frame is privileged
- 3
each measures the other's length as contracted because they disagree about which events are simultaneous, and a length measurement requires locating both ends at the same time
- 4
the symmetry would be broken only if one observer accelerated — an accelerating frame is not inertial, and that observer would feel the acceleration and know it
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