6.2 Simple Harmonic Motion

An experiment is carried out on Planet M using two pieces of equipment. The first piece is a pendulum of length l and the second piece is a block of mass m attached to a spring with spring constant k. The block moves horizontally on a frictionless surface. A motion sensor with a lightbulb is placed above the equilibrium position of the block. Every time the block passes the equilibrium position, the lightbulb lights up.  

Figure 1

The pendulum and the block are both displaced from their equilibrium positions and are made to oscillate with simple harmonic motion. The pendulum bob completes 150 full oscillations in seven minutes and the lightbulb lights up once every 0.70 seconds.  

The mass of the block m is 320 grams. 

Show that the value of k  is approximately 6 N m^-1.

Show that l =

The volume of planet M is the same as the volume of the Earth and the density of Planet M is twice of the density of Earth. Calculate the value of l.

When the acceleration of the pendulum bob is at its maximum, the angle that the string makes with the horizontal is 68°. Find the maximum speed reached by the pendulum bob.

A buoy, floating in a vertical tube, generates energy from the movement of water waves on the surface of the sea. When the buoy moves up, a cable turns a generator on the sea bed producing power. When the buoy moves down, the cable is wound in by a mechanism in the generator and no power is produced. 

Figure 1

The vertical motion of the buoy can be assumed to be simple harmonic. 

A wave of amplitude 3.8 m and wavelength 28 m, moves with a speed of 3.2 m s^–1.

Calculate the maximum vertical speed of the buoy caused by the movement of the wave.

Use the axes provided in Figure 2 to sketch a graph to show the variation with time of the generator output power. Label the time axis with a suitable scale. 

Figure 2

Determining the best location for wave generators, such as the one in Figure 1, is very important. The graph in Figure 3 gives an indication of the relationship between the amplitude of ocean waves and their period. 

Figure 3

Discuss the use of wave generators in terms of the potential electrical power output. In your answer you should: 

• Explain how wave height and wave period affect the energy within a wave
• Describe how different energy losses in the system might affect the power output

A small metal pendulum bob is suspended at rest from a fixed point with a length of thread of negligible mass. Air resistance is negligible. The pendulum begins to oscillate. Figure 1 shows the variation of kinetic energy of the pendulum bob with time.

Figure 1

Figure 2 shows how the kinetic energy of the pendulum varies with displacement.

Figure 2

Write down the equation that models the variation of position with time for the simple harmonic motion of this pendulum.

Calculate the maximum force upon the pendulum when it has a mass of 2.78 kg.

The equation that describes simple harmonic motion is given by 

         a = x  

(i)         State the meaning of each symbol. 

(ii)        Explain the significance of the negative sign.

The bob of a simple pendulum, of mass 35 g, swings with an amplitude of 61 mm. It takes 51.2 s to complete 20 oscillations. 

Calculate the length of the pendulum.

Calculate the magnitude of the restoring force that acts on the bob when at its maximum displacement.

Figure 1 shows a graph of displacement against time for the pendulum. 

Sketch, on Figure 1, graphs of acceleration against time for the pendulum and the total potential energy against time for the pendulum. 

Figure 1

State the energy transfers taking place during one complete oscillation of a vertical mass-spring system, starting when the mass is at its highest point.

A spring, which obeys Hooke’s law, hangs vertically from a fixed support. A force of 3.0 N is required to produce an extension of 60 mm.

A mass of 0.70 kg is attached to the lower end of the spring and is pulled down a distance of 10 mm from the equilibrium position before being released. 

Show that the frequency of the simple harmonic vibrations is 1.3 Hz.

Sketch the displacements of the mass against time on Figure 1, starting from the moment of release and continuing for two full oscillations. Show appropriate time and distance scales on the axes. 

Figure 1

(i)         State at which point in the cycle the mass has its maximum acceleration. 

(ii)        Calculate the acceleration of the mass at that point.

Figure 1 shows one cycle of the displacement-time graphs for two mass-spring systems X and Y that are performing simple harmonic motion. 

Figure 1

The springs used in oscillators X and Y have the same spring constant. 

Using information from Figure 1, determine the relation between the mass used in oscillator Y and that in oscillator X.

Explain briefly how would you use one of the graphs in Figure 1 to confirm that the motion is simple harmonic.

Figure 2 shows how the potential energy of oscillator Y varies with displacement. 

Figure 2

Draw on Figure 2: 

(i)        A graph to show how the kinetic energy of the mass used in oscillator Y varies with its displacement. Label this A. 

(ii)       A graph to show how the kinetic energy of the mass used in oscillator X varies with its displacement. Label this B.

Use data from the graphs to determine the spring constant of the springs used.