The Physics
Hypertextbook
Opus in profectus

# Springs

## Problems

### practice

1. A chest expander is a type of exercise device made from three or four large parallel springs connected to a pair of handles. The springs offer muscle resistance in a way that is similar to lifting weights. One such device broke and was donated to a physics teacher. The teacher attached one of the springs to a ceiling beam in a classroom at started loading it with 1 kg masses. A pair of vertically stacked meter sticks were placed parallel to the loaded spring and the position of the bottom of the spring was recorded. Use this data to determine the spring constant of this spring.
2. A teacher presses down on a spring loaded pop-up toy until the suction cup part of it catches. The teacher's class waits in anticipation. Without warning the suction breaks and the toy does what it's supposed to do — it pops up. The teacher records the toy's mass, its height height before and after being compressed, and the maximum height it attains after popping up.
Pop-up toy
mass
of toy
uncompressed
height
compressed
height
maximum
height
16 g
0.016 kg
11cm
0.11 m
7 cm
0.07 m
92 cm
0.92 m
Assume that the trajectory of the toy was nearly vertical and that air resistance and other forms of friction are negligible and determine the following quantities…
1. the gravitational potential energy of the toy at its highest point relative to the table top
2. the kinetic energy of the toy as it left the table
3. the speed of the toy as it left the table
4. the elastic potential energy of the toy after it was compressed
5. the spring constant of the spring inside the toy
3. Write something different.
4. Write something completely different.

### numerical

1. A spring with a natural height of 57 mm is compressed by a 300 g mass to a new height of 51 mm.
1. Find the spring constant in SI units.
2. Find the height of the spring if the 300 g mass were replaced by a 400 g mass.
2. The graph on the right shows the applied force vs. the extension for a particular spring.
1. Find the spring constant.
2. Find the work done on the spring.
3. Find the potential energy stored in the spring.
3. A spring operated dart gun fires 10 g darts. Arming the gun requires 185 N of force and results in the shortening of the spring by 10 cm.
1. Find the spring constant.
2. Find the energy stored in the spring.
3. Find the muzzle velocity of the dart.
4. If the dart is launched vertically, how high will it rise? (Do not use an equation of motion to solve this problem. Use conservation of energy.)
4. A 68 kg acrobat stands on a spring loaded platform as a part of a stunt. If the spring under the platform has a spring constant of 1.6 × 103 N/m, how far should the spring be compressed to vault the acrobat to a height of 5.0 m?
5. Estimate the elastic potential energy stored in the pole used in the pole vault given that elite vaulters are capable of heights in excess of 6 m and that world class sprinters can cover 100 m in about 10 s. (Assume that a typical male vaulter has a mass of about 80 kg.)
6. An archer puts a 30 g arrow to the bowstring. A force of 200 N is needed to draw the string back 80 cm. When bow is released, 50% of the elastic potential energy is transferred to the arrow.
1. If the arrow is shot horizontally, with what speed does it leave the bow?
2. If the arrow is shot straight up, how high does it rise (assuming air resistance was negligible).
7. The Mary Rose was an English warship that sunk off the coast of Britain in 1545. It was recovered in 1982 and is now on display in the Mary Rose Museum in Portsmouth, England. Read the following passage from a paper on late medieval weapons technology, including longbows recovered from the Mary Rose.

Ottoman Turks in the fourteenth to the sixteenth centuries seem to have used substantially more powerful bows, with draw-weights comparable to strong Mary Rose longbows. However, since these were used with shorter draws and lighter arrows, they still did not have equal terminal effects. A middling 110 lb at 28 in Ottoman bow using a heavy Turkish war arrow might impart to an arrow around 95 joules (70 foot-pounds) of kinetic energy and 2.76 kg m/sec (20 pound feet per second) momentum, whereas a middling 150 lb at 32 in longbow might impart 136 joules (100 foot-pounds) and 5.1 kg m/sec (36.9 pound feet per second), i.e. 43% more kinetic energy and a formidable 89% more momentum.

Clifford J. Rogers, 2011

Using the quantities stated in this passage, compare the Ottoman bow to the English longbow.

1. Which of these late medieval weapons was more efficient?
2. Which used the heavier arrow?
3. Which had the faster arrow?
4. Verify the final claims of the author. Does the English longbow produce 43% more kinetic energy and 89% more momentum than the Ottoman bow?

### algebraic

1. A block of mass m rests on a rough horizontal surface whose coefficient of kinetic friction is μk. The block is held in contact with one end of a spring which is compressed a distance x. The spring has a spring constant of k. The other end of the spring is fixed. If the block is released, over what distance will it travel from its initial position before it comes to rest? Express your answer in terms of the quantities provided in the description above (m, μk, x, k) and the acceleration due to gravity (g).