Simple Harmonic Motion Practice Problems

Fs = -kx

* = We do together. All others you do for homework.

  1. *A hand exerciser utilizes a coiled spring. A force of 89 N is required to compress the spring by 0.0191 m. Determine the force needed to compress the spring by 0.0508 m. (236.71 N)
  2. The graph shows the force that an archer applies to the string of a long bow versus the displacement, x, of the string. Drawing back this bow is analogous to stretching a spring. From the data in the graph, determine the effective spring constant of the bow. (F) (666.67 N/m)

160N

0.24 m (x)

  1. *A block is attached to one end of a spring, k = 100 N/m, hanging from the ceiling. The spring stretches 0.23 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? (0.115 m)
  2. Three identical springs hang from a ceiling. Nothing is attached to the first spring and it hangs down 20cm. A 4.5 N block hangs from the second spring and the end of the spring stretches to 35cm. A block of unknown weight hangs from the 3rd spring and the end of the spring stretches to 50cm. What is
  3. The spring constant of the spring(s)? (30 N/m)
  4. The weight of the unknown block? (9N)
  5. A spring, k = 830 N/m, is hanging from the ceiling of an elevator and a 5 kg mass is attached to the lower end. By how much does the spring stretch (relative to its unstrained length) when the elevator is accelerating up at 0.60 m/s/s? (6.27 E-2 m)
  6. Someone holds a 2.3 kg block pressed against a vertical wall by pressing a horizontally oriented spring (k = 480 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 0.051m. What is the coefficient of static friction? (0.921)
  7. A small ball is attached to one end of a spring that has an unstrained length of 0.20 m. The spring is held by the other end and the ball is whirled around in a horizontal circle at a speed of 3m/s. The spring stretches by 0.010m while it is being whirled. How much would the spring stretch if it were attached to the ceiling and the ball is allowed to hang down motionless? (2.29 E-3 m)
  8. A 30 kg block is resting on a flat horizontal table. On top of this block is resting a 15 kg block to which one end of a horizontal a spring is attached and the other end of the spring is attached to a wall. The spring constant is 325 N/m. The coefficient of kinetic friction between the lower block and the table is 0.6 and the coefficient of static friction between the two blocks is 0.9. A horizontal force is applied to the lower block (pushing it toward the wall) and this force increases in such a way to keep the blocks moving with constant speed. At the point where the upper block begins to slip on the lower block determine the amount by which the spring is compressed. (0.407 m)

The Reference Circle and SHM

  1. Make sure you can read a sin wave graph of x vs t for an object oscillating in SHM (I’ll draw some and quiz you in

class)….for example, determining the spring constant from determining the period (from a graph of its SHM) and its mass.

  1. *The fan blades on a jet engine make one thousand revolutions in a time of 50 ms. Determine
  2. The period (in seconds). (5 E-5 sec/rev)
  3. The frequency (in Hz). (2 E4 rev/sec or Hz)
  4. *The springs in the shock absorbers in the suspension system of a car each support 320kg of the car’s mass. If someone pushes down on the car and notices that it vibrates through 5 cycles in 3 seconds, what is the spring constant of each spring? (3.51 E4 N/m)
  5. A spring stretches by 0.018m when a 2.8 kg object is suspended from its end. How much mass should be attached to this spring so that its frequency is 3 Hz when placed in SHM? (4.29 kg total or 1.49 kg additional mass)

Energy and SHM

  1. *An archer pulls the bowstring back for a distance of 0.470 m before releasing the arrow. The bow and string act like a spring whose spring constant is 425 N/m.
  2. What is the elastic PE of the drawn bow? (46.94 J)
  3. The arrow has a mass of 0.033 kg. How fast is it traveling when it leaves the bow? (53.34 m/s)
  4. A 0.01 kg block is resting on a horizontal frictionless surface and is attached to a horizontal spring whose spring constant is 124 N/m. The block is shoved parallel to the spring axis and set into SHM. Its speed is 8 m/s as it passes through equilibrium. What is the resulting amplitude of the SHM? (7.18 E-2 m)
  5. A heavy-duty staple gun uses a 0.150 kg metal rod that rams against the staple to eject it. The rod is pushed by a spring, k = 34,000 N/m. Squeezing the handle of the gun compresses the spring by 3.5 E-2 m from its unstrained length and then releases it. If the ram and spring are oriented vertically and the spring is still compressed by 1 E –2 m when the downward moving ram hits the staple, find the speed of the ram at the instant of contact. (15.98 m/s)
  6. *A 2 kg block is hanging from the end of a vertical spring of spring constant 50 N/m. The object is pulled 0.20 m down from the equilibrium point and released from rest (we’ll call h = 0 m at the equilibrium point). Taking equilibrium to be the point where the spring has zero elastic potential energy (it doesn’t, but set it to be true for simplicity), what is the
  7. KE at 0 m, -0.2 m, and +0.2 m? (1 J, 0 J, 0J)
  8. Elastic PE at the same heights? (0J, 1 J, 1 J)
  9. A 1.1 kg object is suspended from a vertical spring, k = 120 N/m.
  10. How much is the spring stretched from its unstrained length? (8.98 E-2 m)
  11. The object is pulled down an additional 0.2 m and released from rest. What speed does the object have as it passes through the equilibrium point? (2.09 m/s)
  12. *A 3.5 kg wooden block moving at 5 m/s on a frictionless horizontal surface strikes one end of a spring (600 N/m) whose other end is attached to a wall.
  13. How much work did the block do on the spring? (43.75 J)
  14. Is the work done by the block conservative or non-conservative? Why?
  15. What is the maximum displacement of the spring? (0.38 m)
  16. A 6 kg block is attached to one end of a spring (k= 125 N/m) on a frictionless horizontal surface. The other end of the spring is attached to a wall and the spring is compressed 7 cm.
  17. How much work was needed to compress the spring to this amount? (0.306 J)
  18. What is the final velocity of the block after the mass is released? (0.320 J)
  19. *A 5 kg block and a 4 kg block are held in place at opposite ends of a compressed spring (x = 0.23 m, k = 400 N/m) along a frictionless horizontal surface.
  20. How much total energy does the system have before it is released? (10.58 J)
  21. What is the initial momentum of each block before the system is released? (0 N m)
  22. When the blocks are released from rest (and the spring is compressed by 0.23 m), what is the maximum speed of each block? (1.37 and 1.71 m/s, opposite in direction from each other)

The Pendulum

  1. *If the period of a simple pendulum is to be 2 s, what should be its length? (0.993 m)
  2. A spiral staircase winds up to the top of a tower in an old castle. To measure the height of the tower, a rope is attached to the top of the tower and hung down the center of the staircase. However, nothing is available with which to measure the length of the rope. So, at the bottom of the rope a small object is attached so as to form a simple pendulum that just clears the floor. The period of the pendulum is measured to be 9.2 s. What is the height of the tower? (21.01 m)
  3. A pendulum clock can be approximated as a simple pendulum of length 1.00 m and keeps accurate time at a location where g = 9.83 m/s/s. In a location where g = 9.78 m/s/s, what must be the new length of the pendulum such that the clock continues to keep accurate time (the period remains the same)? (0.995 m)
  4. Astronauts on a distant planet set up a simple pendulum of length 1.2 m. The pendulum executes SHM and makes 100 complete vibrations in 280 seconds. What is the acceleration due to gravity at the location? (6.04 m/s/s)
  5. A simple pendulum is made from a 0.65 m long string and a small ball attached to its free end. The ball is pulled to one side through a small angle and then released from rest. After the ball is released, how much time elapses before it attains its greatest speed? (0.405 s)

Fluids Practice Problems

FB = ρfluidVfluid g 1 atm = 1 x 105 N/m2 = 1 x 105 Pa

A1v1 = A2v2

Mass Densities Table of Common Substances

Substance / ρ Mass Density (kg/m3)
Solids
Aluminum / 2, 700
Brass / 8, 470
Concrete / 2, 200
Copper / 8, 890
Diamond / 3, 520
Gold / 19, 300
Ice / 917
Iron (steel) / 7, 860
Lead / 11, 300
Quartz / 2, 660
Silver / 10, 500
Wood (yellow pine) / 550
Liquids
Blood (whole, 370 C) / 1, 060
Ethyl alcohol / 806
Mercury / 13, 600
Oil (hydraulic) / 800
Water (40 C) / 1, 000
Gases
Air / 1.29
Carbon Dioxide / 1.98
Helium / 0.179
Hydrogen / 0.0899
Nitrogen / 1.25
Oxygen / 1.43
  1. *The ice on a lake is 0.01 m thick. The lake is circular with a radius of 480 m. Find the mass of the ice. (V = π r2 h) (6.64 E6 Kg)
  2. One end of a wire is attached to a ceiling and a solid brass ball is tied to the lower end. The tension in the wire is 120 N. What is the radius of the brass ball? (Vsphere = 4/3 π r3) (7.01 E-2 m)
  3. The body of a man whose weight is about 690 N contains about 5.2 E-3 m3 of blood.
  4. Find the blood’s weight. (54.02 N)
  5. Express it as a percentage of the body weight. (7.83 %)
  6. *Suppose the pressure acting on the back of a swimmer’s hand is 1.2 E5 Pa at the bottom of a swimming pool. The surface area of the back of the hand is 8.4 E-3 m2. What is the magnitude of the force that acts on the hand due to the water? (1008 N)
  7. 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 x10-4 m2, what is the pressure exerted on the piston by the gas? Be careful here… (3.14 E 5 Pa)
  1. *What is the pressure at a point 5.5 m beneath the surface of a lake? Compare this to atmospheric pressure. (1.54 E 5 Pa)
  1. The Mindanao Trench, off the coast of the Philippines, is the world's deepest at 11,515 m. The density of seawater is 1025 kg/m3.
  2. What is the pressure at this depth? (1.16 E 8 Pa)
  3. If an underwater submersible were to explore this depth, what force would the water exert on the circular observation window (radius of 0.10 m)? Compare this force to the weight of a jet liner, mass of 1.2 E5 Kg. (3.64 E 6 N, ~ double the weight of the airliner just on that little area---that’s a lot!)
  4. If the inside of the submersible were kept at atmospheric pressure, what is the net force on the window? (3.64 E 6 N; the inside pressure is soo small it doesn’t make a dent in equalizing the total pressure)
  5. If the heart is 1.35m above the Anterior Tibial artery in the foot, how much does the blood pressure in the foot exceed that in the heart? (1.40 E 4 Pa)
  1. In a hydraulic car lift, the input piston has a radius of 0.0120 and the output plunger has a radius of 0.150 m. If a car sits on the output plunger (with a combined car/plunger weight of 20,500 N), what input force is needed to just support the car and the output plunger when the bottoms of the plungers are at the same height? (131.20 N)

Archimedes Principal

  1. *A 0.1m x 0.2 m x 0.3 m block is completely submerged underwater.
  2. What is the buoyant force on the block? (58.8 N)
  3. If the block is made of yellow pine wood, what is the net force on the block? (26.46 N, up)
  4. *A solid, square pine wood raft measures 4m on a side and is 0.30 m thick.
  5. Does the raft float in water? (yes, FB (max) > mg or the density of pine is less than the density of water)
  6. If so, how much of the raft is beneath the surface? (0.165 m)
  7. A Goodyear blimp contains about 5.40 E3 m3 of Helium (density of 0.179 kg/m3). Find the maximum weight of the load (including the materials that make up the inflated balloon) that the blimp can carry in equilibrium and at an altitude where the density of air of 1.2 kg/m3. (6.35 E 4 N)
  8. A father (830 N) and his smaller daughter (340N) are sitting on two different size beach balls that are just submerged beneath the water. Ignoring the weight of the air inside the beach balls and the parts of their legs beneath the water, find the radius of each ball. (Vsphere = 4/3 π r3) (0.27 m, 0.20 m)
  9. A bodybuilder is holding a 29 kg steel barbell above her head. How much force would she have to exert if the barbell were lifted underwater? (248.05 N)
  10. A large balloon, filled with an unknown gas, exerts a 5.25 N upward force on a spring (that is vertically oriented and solidly connected to a table---remember Newton’s 3rd Law here). If the volume of the balloon is 1.70 m3, what is the density of the gas? Take the density of air to be 1.21 kg/m3. (0.89 Kg/m3)

*Note: A spring scale/spring balance is just a device that measures the force pulling on the spring inside the instrument. It consists of a spring with a calibrated spring constant that will give a reading in Newtons of the amount of force pushing or pulling on it.

  1. A class conducts a number of experiments on the behavior of fluids using an instructor, a rope, a force scale, and a very large graduated cylinder. The cylinder is 3.00 m deep and has a diameter of 2.00 m. The 80-kg instructor is completely immersed in the water in the cylinder and tied to a spring scale at the bottom of the cylinder as shown in the diagram below. The water in the cylinder rises 2.65 cmafter the instructor is immersed. Vcylinder =  r2 h
  2. How dense is the instructor? (960.94 kg/m3)
  3. What is the reading on the spring scale? (31.85 N)

Equation of Continuity and Bernoulli’s Equation

  1. *Water enters a pipe of diameter 3.0 cm with a velocity of 3.0 m/s. The water encounters a constriction and its velocity changes to 12 m/s. What is the diameter of the constricted portion of the pipe? (0.015 m)
  2. *An aneurysm is an abnormal enlargement of a blood vessel such as the aorta. Suppose that, because of an aneurysm, the cross-sectional area (A1) increases to a value of A2= 1.71A1. The speed of the blood (density = 1060 kg/m3) through a normal portion of the aorta is v1= 0.40 m/s. If the person is lying down (so the aorta is horizontal), determine the amount the pressure in the enlarged area (P2) exceeds the pressure in the normal region (P1). (55.80 Pa)
  3. 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? (-3.92 E4 Pa)
  1. 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)
  1. A truck with a tarp covering the cargo area is moving down the highway at 27 m/s. If the density of air is 1.29 kg/m3, how much does the pressure inside the cargo area beneath the tarp exceed the outside pressure? (470.21 Pa)
  2. An airplane wing is designed so that the speed of the air across the top of the wing is 251 m/s when the speed below the wing is 225 m/s. The density of the air is 1.29 kg/m3. What is the net upward force on a wing with an area of 24.0 m2? (1.92 E 5 N)

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Unit 5 Practice ProblemsSimple Harmonic Motion and Fluids