9.1 Simple Harmonic Motion

Question 1b

Marks: 2

The ball is oscillating in simple harmonic motion. The graph shows the displacement of the ball over time.

9-1-hl-sq-medium-1b-s-t-graph-question-stem

(b)

Determine the maximum velocity of the ball.

[2]

Key Concepts

Question 1c

Marks: 3

(c)

For the motion of the ball

(i)

Show that the acceleration of the ball at 90 ms is 43 m s⁻².

[2]

(ii)

On the graph, mark an X at a point where the resultant force acting on the ball is zero.

[1]

Key Concepts

Question 1d

Marks: 3

(d)

On the axes provided, taking the oscillation as beginning at the positive amplitude when t = 0, sketch for the ball's motion:

(i)

The variation of displacement with time, label this graph x

[1]

(ii)

The variation of velocity with time, label this graph v

[1]

(iii)

The variation of acceleration with time, label this graph a

[1]

9-1-hl-sq-medium-1d-question-stem-1

Key Concepts

Question 2a

Marks: 4

A mass of 45 g on a spring undergoes simple harmonic motion with a period of 0.84 s.

A wooden block attached to the same spring undergoes simple harmonic motion with a period of 0.64 s. The wooden block is displaced horizontally by 3.6 cm from the equilibrium position on a frictionless surface.

9-1-hl-sq-medium-2a-diagram

(a)

Determine the total energy in the oscillation of the wooden block.

[4]

Key Concepts

Question 2b

Marks: 3

(b)

Using the information from part (a), sketch on the axes the kinetic, potential, and total energies of the oscillating wooden block as they vary with displacement.

[3]

Key Concepts

Question 2c

Marks: 3

The investigation from part (a) is repeated. The same wooden block and spring are used, but a second identical spring is added in parallel.

9-1-hl-sq-medium-2c-diagram

(c)

Suggest how this change will affect the fractional uncertainty in the mass of the wooden block.

[3]

Key Concepts

Question 2d

Marks: 3

The spring from part (a) is attached to the mass in part (a), and oscillates freely in simple harmonic motion.

The graph shows the variation of displacement with time t of the mass on the spring.

9-1-hl-sq-medium-2d-diagram

(d)

For the new mass-spring system

(i)

Describe the motion of the mass on the spring.

[1]

(ii)

Determine the initial energy of the mass-spring system.

[2]

Key Concepts

Question 3a

Marks: 2

A mass of 75 g is connected between two identical springs. The mass-spring system rests on a frictionless surface. A force of 0.025 N is needed to compress or extend the spring by 1.0 mm.

The mass is pulled from its equilibrium position to the right by 0.055 m and then released. The mass oscillates about the equilibrium position in simple harmonic motion.

9-1-hl-sq-medium-3a-mass-spring

The mass-spring system can be used to model the motion of an ion in a crystal lattice structure.

9-1-hl-sq-medium-3a-ion

The frequency of the oscillation of the ion is 8 × 10¹² Hz and the mass of the ion is 6 × 10⁻²⁶ kg. The amplitude of the vibration of the ion is 2 × 10⁻¹¹ m.

(a)

For the oscillations

(i) Calculate the acceleration of the mass at the moment of release.

[1]

(ii) Estimate the maximum kinetic energy of the ion.

[1]

Key Concepts

Question 3b

Marks: 3

(b)

For the mass-spring system

(i)

Calculate the total energy of the system.

[1]

(ii)

Use the axes to sketch a graph showing the variation over time of the kinetic energy of the mass and the potential energy of the springs.

You should include appropriate values, and show the oscillation over one full period.

9-1-hl-sq-medium-3b-axes

[2]

Key Concepts

Question 3c

Marks: 3

The same mass and a single spring from part (a) are attached to a rigid horizontal support.

9-1-hl-sq-medium-3c-vertical-spring

The length of the spring with the mass attached is 64 mm. The mass is pulled downwards until the length of the spring is 76 mm. The mass is released, and the vertical mass-spring system performs simple harmonic motion.

(c)

For the new mass-spring system

(i)

Determine the velocity of the mass 2 seconds after its release.

[1]

(ii)

Determine the kinetic energy of the mass at this point.

[2]

Key Concepts

Question 3d

Marks: 2

The diagram shows the vertical spring-mass system as it moves through one period.

9-1-hl-sq-medium-3d-diagram

(d)

Annotate the diagram to show when:

E_p = max

E_k = max

v = 0

v = max

[2]

Key Concepts

Question 4a

Marks: 2

A ball is displaced through a small distance x from the bottom of a bowl and then released.

9-1-hl-sq-medium-4a-diagram

The frequency of the resulting oscillation is 1.5 Hz and the maximum velocity reaches 0.36 m s⁻¹. r is the radius of the bowl.

(a)

For the oscillating ball:

(i)

Show that the radius of the bowl in which it oscillates is approximately 11 cm.

[1]

(ii)

Calculate the amplitude of oscillation.

[1]

Key Concepts

Question 4b

Marks: 3

(b)

For the ball

(i)

Calculate the time taken for the displacement to fall to 0.01 m after it is released.

[2]

(ii)

Determine the acceleration at 0.2 s after it is released.

[1]

Key Concepts

Question 4c

Marks: 4

(c)

Sketch the graphs showing how the displacement, velocity and acceleration of the ball vary with time. You should include any relevant values.

9-1-hl-sq-medium-4c-axes

[4]

Key Concepts

Question 4d

Marks: 2

The ball was replaced by a ball of the same size, but with a greater mass.

(d)

Outline what effect this would have on the period of the oscillation.

[2]

Key Concepts

Question 5a

Marks: 2

An experiment is carried out on Planet Z using a simple pendulum and a mass-spring system. The block moves horizontally on a frictionless surface. A motion sensor is placed above the equilibrium position of the block which lights up every time the block passes it.

q2b-figure-1

The pendulum and the block are displaced from their equilibrium positions and oscillate with simple harmonic motion. The pendulum bob completes 150 full oscillations in seven minutes and the bulb lights up once every 0.70 seconds. The block has a mass of 349 g.

(a)

Show that the value of the spring constant k is approximately 7 N m⁻¹.

[2]

Key Concepts

Question 5b

Marks: 4

The volume of Planet Z is the same as the volume of Earth, but Planet Z is twice as dense.

(b)

For the experiment on Planet Z

(i)

Show that the length of the pendulum,

l = \frac{4 m g}{k}

[2]

(ii)

Calculate the value of $l$ .

[2]

Key Concepts

Question 5c

Marks: 3

The angle that the pendulum string makes with the horizontal is 81.4 ° when the acceleration of the pendulum bob is at a maximum.

(c)

Determine the maximum speed reached my the pendulum bob.

[3]

Key Concepts

Question 5d

Marks: 2

(d)

Compare and contrast how performing the experiment on Planet Z, rather than on Earth, affects the period of the oscillations of the pendulum and the mass-spring system.

[2]

Key Concepts