10 points, suggested time 20 minutes
Two students investigate how acceleration depends on mass distribution in a two-mass Atwood system. They set up the apparatus shown in Figure 1, with a block of mass m on the left and a block of mass 2m on the right, connected by a light, inextensible string passing over a low-friction, massless pulley. The students release the blocks from rest and observe the motion. Air resistance is negligible.
Part (a)
Starting with Newton’s second law, derive an expression for the magnitude of the acceleration a of the blocks. Express your answer in terms of m and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Part (b)
Derive an expression for the tension T in the string in terms of m and physical constants, as appropriate.
Part (c)
Derive an expression for the speed v of the 2m block after it has fallen a distance h, in terms of h and physical constants, as appropriate.
Part (d)
The students modify the setup so the lighter block is replaced with a block of mass 3m while the heavier block remains at mass 2m.
i. Derive an expression for the new acceleration a′ of the system in terms of g.
ii. Express the percent change in magnitude of the acceleration compared to Part (a), as a positive or negative percentage.
iii. Indicate whether the direction of acceleration reverses, and justify briefly.
Answer all parts of Question 1. Show your work for each part, including derivations and justifications where required.
12 points, suggested time 25 minutes
A student investigates the motion of a 2.0 kg block on a horizontal, frictionless track as a variable horizontal force Fx(t) is applied. Using a force sensor and motion detector, the student records the force-versus-time graph shown in Figure 1, with the block initially at rest at t = 0. The graph consists of three piecewise-constant segments:
Part (a)
On a set of axes with px (kg·m/s) on the vertical axis and t (s) on the horizontal axis, sketch and label a graph of the x-component of the momentum of the block from t = 0 to t = 7 s. Include the value of px at every segment endpoint.
Note: This interface accepts text responses only. Describe your figure in precise words — axis labels with units, key coordinate points, slopes, intercepts, arrows with labels. Your written description is graded against the same rubric as a drawn figure.
Part (b)
Calculate the speed of the block at t = 7 s.
Part (c)
Indicate the time interval during which the kinetic energy of the block is at its MAXIMUM value, and calculate that maximum value. Justify your answer by referencing the px vs. t graph you drew in Part (a).
Part (d)
Calculate the total impulse delivered to the block from t = 0 to t= 7 s. Express your answer in N·s with direction.
Answer all parts of Question 2. Show your work for each part, including calculations and justifications where required.