Simple Harmonic Motion and Fluids Review Problems
Springs
1. Gina's favorite exercise equipment at the gym consists of various springs. In one exercise, she pulls a handle grip attached to the free end of a spring to 0.80 m from its unstrained position. The other end of the spring (spring constant = 53 N/m) is held in place by the equipment frame. What is the magnitude of the force that Gina is applying to the handle grip? (42.4 N)
2. A 25-coil spring with a spring constant of 350 N/m is cut into five equal springs with five coils each. What is the spring constant of each of the 5-coil springs? (1750 N/m)
3. A block is suspended from the ceiling by a long, thin strip of tungsten metal. The strip behaves as an ideal spring with horizontal deflections. To produce a 0.25 m horizontal deflection of the block, a force of 6.5 N is required. Calculate the spring constant for the tungsten strip. (26 N/m)
4. A spring has a constant of 248 N/m. Find the magnitude of the force needed to
a. Stretch the spring by 3 E –2 m from its unstrained length. (7.44 N)
b. Compress the spring by the same amount. (7.44 N)
c. How much work is need to stretch or compress the spring by the above amounts? (0.11 N)
5. A block is attached to one end of a spring, k = 200 N/m, hanging from the ceiling. The spring stretches 0.65 m. If the spring is cut exactly in half and the same block is suspended from the end of one of the sections, how far will the spring stretch now? (0.325 m)
6. Someone holds a 3 kg block pressed against a vertical wall by pressing a horizontally oriented spring (k = 500 N/m) that is sandwiched between their hand and the block. They press on the spring until the block stops slipping down the wall. At this point the spring is compressed by 0.075m. What is the coefficient of static friction? (0.784)
7. A spring stretches by 0.023 m when a 3 kg object is suspended from its end. How much mass should be attached to this spring so that its frequency is 4 Hz? (2.02 kg)
8. A 0.750-kg object hanging from a vertical spring is observed to oscillate with a period of 2.00 s in SHM. When the 0.750-kg object is removed and replaced by a 1.25-kg object, what will be the period of oscillation if set in SHM? (2.58 s)
9. A certain spring compressed 0.20 m has 10 J of elastic potential energy. The spring is then cut into two equal halves and one of the halves is compressed by 0.20 m. How much potential energy is stored in the compressed half of the spring? (20 J)
10. A 0.2-kg block is held in place by a force F that results in a 0.10-m compression of a spring beneath the block. The spring constant is 100 N/m. Assuming the mass of the spring is negligible compared to that of the block, to what maximum height would the block rise if the force F were removed? (0.26 m)
11. A relaxed spring protrudes from an opening 0.050 meters as shown in Figure A. When a 1.00-kg block is placed on the spring, the spring compresses and the block and spring eventually reach equilibrium. At the equilibrium point, the block is at ground level, but not in contact with the ground as shown in Figure B.
a. Determine the spring constant k. (196N/m)
b. How much potential energy is stored in the spring in Figure B? (0.245 J)
12. A spring with constant k = 60.0 N/m is at the base of a frictionless, 40.0°-inclined plane. A 0.70-kg block is a distance of 0.89m up the incline. When the block is released from rest, it slides the 0.89 m down the incline, hitting the spring and compressing it to eventually come to a stop. How much was the spring compressed from equilibrium? (0.36 m)
13. A 7 kg block and a 5 kg block are held in place at opposite ends of a compressed spring (k = 500 N/m) along a frictionless horizontal surface. When the blocks are released from rest (and the spring is compressed by 0.40 m), what is the maximum speed of each block? (2.18 m/s and 3.06 m/s, respectively)
Pendulums
14. What is the period of pendulum consisting of a 6-kg object oscillating on a 4-m string? (4.01 s)
15. A pendulum is transported from sea-level, where the acceleration due to gravity g = 9.80 m/s2, to the bottom of Death Valley. At this location, the period of the pendulum is decreased by 3.00%. What is the value of g in Death Valley? (10.42 m/s)…this is just a lot of algebra….
FLUIDS
16. What mass of water (at 4.0 °C) can be contained in a rectangular box whose dimensions are 10.0 cm by 5.00 cm by 1.00 cm? The density of water at 4.0 °C is 1.000 ´ 103 kg/m3. (0.05 kg)
17. The average density of the material in intergalactic space is approximately 2.5 ´ 10-27 kg/m3. What is the volume of a gold sample, r = 19 300 kg/m3, that has the same mass as 8.0 ´ 1024 m3 of intergalactic space? (1.04 E -6 m3)
18. Within a certain type of star called a neutron star, the material at the center has a mass density of 1.0 ´ 1018 kg/m3. If a small sphere of this material of radius 1.0 ´ 10-5 m were somehow transported to the surface of the earth, what would be the weight of this sphere? (4.11 E 4 N)
19. A solid cylinder has a radius of 0.051 m and a height of 0.0030 m. The cylinder is composed of two different materials with mass densities of 1950 kg/m3 and 1470 kg/m3. If each of the two materials occupies an equal volume, what is the total mass of the cylinder? (4.19 E-2 kg)
20. A gas sample is confined to a chamber with a movable piston. A small load (open to the atmosphere) is placed on the piston and the system is allowed to reach equilibrium. If the weight of the piston and the load is 90.0 N and the piston has an area of 4.2 x10-4 m2, what is the pressure exerted on the piston by the gas? (3.14 E5 Pa)
21. A submarine is operating at 100.0 m below the surface of the ocean. If the air inside the submarine is maintained at atmospheric pressure, what is the magnitude of the net force that acts on the rectangular hatch 2.0 m ´ 1.0 m on the deck of the submarine? Take the density of sea water to be 1025 kg/m3. (2.01 E6 N)
22. In the hydraulic press used in a trash compactor, the radii of the input piston and the output plunger are 6.4 E-3 m and 5.1 E –2 m, respectively. The height difference between the two can be ignored. What force is applied to the trash when the input force is 330N? (2.10 E4 N)
23. A dentist’s chair with a patient in it weighs 2100N. The output plunger of the hydraulic system begins to lift the chair when the dentist’s foot applies a force of 55N to the input piston. Neglecting any height difference between the plunger and piston, what is the ratio of the radii of the plunger to the piston? (6.18)
24. A 2-kg block displaces 10 kg of water when it is held fully immersed. The object is then tied down as shown in the figure; and it displaces 5 kg of water. What is the tension in the string? (29.4 N)
25. A paperweight when weighed in air has a weight of 6.9 N but has a weight in water of 4.3 N when it is completely immersed. What is the volume of the paperweight? (2.65 E-4 m3)
26. Take the density of ice to be 917 kg/m3 and the density of seawater to be 1030 kg/m3. A large iceberg floats in Arctic waters. What fraction of the volume of the iceberg is exposed? (89% underwater or 11% exposed)
27. An object weighs 15 N in air and 13 N when submerged in water. Determine the density of the object. (7.47 E3 kg/m3)
28. After a cargo van drives onto a river ferry, the ferry sinks by 0.05 m. The length and width of the ferry are 20 m and 7 m, respectfully. Determine the weight of the cargo van. (6.86 E4 N)
29. When a block of volume 1.00 ´ 10-3 m3 is hung from a spring scale and is completely submerged in water as shown in Figure A, the scale reads 10.0 N. When the same block is then placed in an unknown liquid, it floats with 2/3 of its volume submerged as suggested in Figure B.
a. Determine the mass of the block. (2.02 kg)
b. Determine the density of the unknown liquid. (3030 kg/m3)
30. Ann uses a hose to water her garden. The water enters the hose through a faucet with a 6.0-cm diameter. The speed of the water at the faucet is 5 m/s. If the faucet and the nozzle are at the same height, and the water leaves the nozzle with a speed of 20 m/s, what is the diameter of the nozzle? (0.03 m)
31. Suppose a 15 m/s wind is blowing across the roof of your house. The density of air is 1.29 Kg/m3.
a. Determine the reduction in (difference in) air pressure (below atmospheric pressure of stationary air) outside the house. (145.13 Pa)
b. Explain why some roofs blow off during high winds.
32. Water is flowing through a pipeline at a constant speed when it encounters a vertical bend in the pipe raising it 4.0 m. The cross sectional area of the pipe does not change. What is the difference in pressure (PB - PA) in the portions of the pipe before and after the rise? (-39,200 Pa)
33. The density of the liquid flowing through the horizontal pipe in the drawing is 800 kg/m3. The pressure of the fluid at point A is 2.5 E5 Pa while at point B it is 1.8 E5 Pa. If the fluid is moving at 6 m/s through A, how fast is it moving through B? (A is the same height as B). (14.53 m/s)
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Unit 5 Review Problems SHM and Fluids