Package Chute with a Spring Bumper
A team is validating a package chute before installation. A 2.40 kg box is released from rest and slides 3.20 m down a straight ramp inclined at 28.0° above horizontal. The manufacturer specifies μk=0.22 between box and ramp, but the ramp has a patch of slightly rougher paint near the lower end. At the bottom, a photogate records 3.90 m/s. The box then compresses a horizontal bumper spring of stiffness 850 N/m; a ruler beside the spring records a maximum compression of 0.205 m. The spring is mounted on a low-friction carriage, so some of the impact may go into moving the carriage rather than compressing the spring. The team must decide whether the chute meets a requirement that the box stop within 0.250 m of bumper travel and whether the discrepancy from the nominal friction model is large enough to require a retest. Assume g=9.81 m/s² and ignore air resistance.
1. Calculate the speed at the bottom predicted by the stated friction coefficient, and compare it with the photogate value. What does the comparison imply about the friction model?
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The vertical drop is h = 3.20 sin 28.0° = 1.50 m. The initial gravitational potential energy is mgh = 2.40 × 9.81 × 1.50 = 35.3 J. The friction force has magnitude μk mg cosθ = 0.22 × 2.40 × 9.81 × cos 28.0° = 4.57 N, so friction does 4.57 × 3.20 = 14.6 J of negative work. The predicted kinetic energy at the bottom is therefore 35.3 − 14.6 = 20.7 J, giving v = sqrt(2 × 20.7 / 2.40) = 4.15 m/s.
The photogate value is 3.90 m/s, whose kinetic energy is 0.5 × 2.40 × 3.90² = 18.25 J. The shortfall is about 2.45 J, or roughly 12% of the predicted kinetic energy. That is too large to dismiss as rounding of g or the angle. It implies the effective friction is greater than the nominal 0.22, for example because of the rougher patch, or that another loss mechanism is present. The nominal μk should not be used as a precise description of this run.
2. Using the observed speed, estimate the maximum spring compression if all kinetic energy is converted into spring potential energy. Does the chute meet the 0.250 m requirement? Would the nominal prediction also meet it?
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If all kinetic energy at impact goes into the spring, then 0.5 mv² = 0.5 kx², so x = sqrt(mv²/k). With v = 3.90 m/s, x = sqrt(2.40 × 3.90² / 850) = sqrt(36.50 / 850) = 0.207 m.
The measured compression is 0.205 m, very close to this value. That suggests most of the impact energy did go into the spring, and any carriage motion or impact loss was small. The requirement of 0.250 m is met with about 0.045 m of margin.
Using the nominal predicted speed of 4.15 m/s gives x = sqrt(2.40 × 4.15² / 850) = 0.220 m. That also meets the requirement, but with only 0.030 m of margin. The chute passes the stopping requirement on both the measured and nominal estimates, though the friction discrepancy should be noted because a higher release speed or a heavier box could consume that margin.
3. The team has one day to decide whether to retest. What measurement or test would best separate extra ramp friction from spring-carriage energy loss, and how would the result affect the decision?
Read the worked answer
Separate the two effects by measuring the box before and during the impact, not just at one photogate. For the ramp, place two gates a known distance apart on the lower section and calculate the acceleration there. If the rough patch is responsible, the local acceleration will be lower than g(sinθ − μk cosθ) for μk = 0.22, and the speed deficit will appear before the bumper. A motion sensor along the ramp would show the same thing.
For the spring, record both spring compression and carriage displacement with a video or a second ruler. Lock the carriage, if it is safe to do so, and repeat the run; if compression rises to the value predicted from the impact speed, the carriage was taking energy. Also check the spring constant with a static load, because a wrong k would shift the compression estimate directly.
The decision should turn on the worst credible case. If the rough patch explains the speed deficit and the measured compression stays below 0.250 m even at the maximum allowed release height, the chute can pass with a note. If either test shows that the current margin could be consumed at the maximum release condition, retest before installation.