IB Physics HL C.1 Simple Harmonic Motion

[Maximum number: 5]

(a)

One end of a light spring is attached to a rigid horizontal support.

An object W of mass 0.15 kg is suspended from the other end of the spring. The extension x of the spring is proportional to the force F causing the extension. The force per unit extension of the spring k is $18 \mathrm{Nm}^{-1}$ .

A student pulls W down such that the extension of the spring increases by 0.040 m . The student releases W and as a result W performs simple harmonic motion (SHM).

[ 5 ]

(i)

State what is meant by the expression "W performs SHM".

[ 2 ]

(ii)

Determine the period of oscillation of the spring.

[ 3 ]

$\omega=\sqrt{\frac{a}{x}}$ ;
$=10.95 \mathrm{~rad} \mathrm{~s}^{-2}$ ;
$T=\left(\frac{2\pi}{\omega}=\right) \frac{6.28}{10.95}=0.57 \mathrm{~s}$ ;

[Maximum number: 15]

This question is in two parts. Part 1 is about simple harmonic motion (SHM). Part 2 is about gas in an engine.

Part 1 Simple harmonic motion (SHM)
An object is placed on a frictionless surface. The object is attached by a spring fixed at one end and oscillates at the end of the spring with simple harmonic motion (SHM).

The tension F in the spring is given by F=k x where x is the extension of the spring and k is a constant.

(a)

Show that $\omega^{2}=\frac{k}{m}$ .

[ 2 ]

m a=-k x;
$a=-\frac{k}{m} x ;($ condone lack of negative sign )

\left(\omega^{2}=\frac{k}{m}\right)

or
implied use of defining equation for simple harmonic motion $a=-\omega^{2} x$ ;
(so $\omega^{2}=\frac{k}{m}$ )
m a=-k x so $a=-\left(\frac{k}{m}\right) x$ ;

(b)

One cycle of the variation of displacement with time is shown for two separate mass-spring systems, A and B .

[ 5 ]

(i)

Calculate the frequency of the oscillation of A.

[ 1 ]

$0.833(\mathrm{~Hz})$ ;

(ii)

The springs used in A and B are identical. Show that the mass in A is equal to the mass in B.

[ 2 ]

frequency/period is the same so $\omega$ is the same;
k is the same (as springs are identical);
(so m is the same)

(iii)

Outline how you would use the graph to confirm that A is performing simple harmonic motion.

[ 2 ]

defines simple harmonic motion as acceleration proportional to negative displacement $/ x=x_{0} \sin \omega t$ ;
correct method for showing how graph leads to the definition;

(c)

The graph shows the variation of the potential energy of A with displacement.

On the axes,

[ 5 ]

(i)

draw a graph to show the variation of kinetic energy with displacement for the mass in A. Label this A.

[ 2 ]

(ii)

sketch a graph to show the variation of kinetic energy with displacement for the mass in B. Label this B.

[ 3 ]

end displacements correct $\pm 0.01 \mathrm{~m}$ ;
maximum lower than 0.16 J ;
maximum equal to $0.04 \mathrm{~J} \pm$ half square;

(d)

Using data from (b) and (c), calculate the mass in A.

[ 3 ]

maximum speed of oscillator $\left.\left(=\omega x_{0}=2 \pi \times 0.833 \times 0.2\right)=1.05\left(\mathrm{~m} \mathrm{~s}^{-1}\right) ;\right\} \begin{aligned} & (\boldsymbol{N} . \boldsymbol{B} . \\ & \left.\omega=5.24 \mathrm{rad} \mathrm{s}^{-1}\right)\end{aligned}$
maximum energy read-off $=0.16(\mathrm{~J}) ;\} \begin{aligned} & \text { (condone ECF for incorrectly drawn } \\ & \text { maximum kinetic energy value in (c)(i)) }\end{aligned}$
use to $\frac{1}{2} m v^{2}$ to give mass value ( $=0.292 \mathrm{~kg}$ using correct data);
or
read-off from graph of $\mathrm{ke}_{\text {max }}=0.16(\mathrm{~J})$ and $x_{\text {max }}=0.2(\mathrm{~m})$ ; (both needed)
$0.16=\frac{1}{2} m \times(2 \pi f)^{2} \times(0.2)^{2} ;$
0.292 (kg); (use of 0.8 Hz gives 0.32)

Allow any alternative method if correct, eg: T from graph, k evaluated $\left(8 \mathrm{Nm}^{-1}\right)$ .

Part 2 Gas in an engine

[Maximum number: 9]

This question is in two parts.

(a)

A particle P moves with simple harmonic motion.

[ 2 ]

(i)

State, with reference to the motion of P , what is meant by simple harmonic motion.

[ 2 ]

(b)

The graph shows how the velocity v of particle P varies with time t.

Use the graph opposite to determine for the motion of P the

[ 7 ]

(i)

period.

[ 1 ]

(ii)

amplitude.

[ 4 ]

Marking guidance:

max velocity $=0.74( \pm 0.02) \mathrm{ms}^{-1}$ ;
recognize max velocity $=\omega x_{0}$ ;
$\omega=\left(\frac{2 \pi}{T}=\frac{2 \pi}{0.30}=\right) 20.9 \mathrm{rads}^{-1} ;$ $x_{0}=\left(\frac{0.74}{20.9}=\right) 3.5( \pm 0.2) \times 10^{-2} \mathrm{~m} ;$
or
identifies displacement with area; uses one quarter of a cycle;
answer in the range of 30 to 40 mm ;
answer in the range of 33 to 37 mm ;

(iii)

displacement of P from equilibrium at $t=0.2 \mathrm{~s}$ .

[ 2 ]

$\quad v=0.64( \pm 0.2) \mathrm{ms}^{-1}$ ;
use $v=\omega \sqrt{\left(x_{0}^{2}-x^{2}\right)}$ to get $x=1.7( \pm 0.2) \times 10^{-2} \mathrm{~m}$ ;
or
recognition that $x=x_{0} \cos \omega t$ ;
$x\left(=35 \cos \left[\frac{2 \pi}{0.3} \times 0.2\right]\right)=17.5 \mathrm{~mm} ;$

[Maximum number: 2]

One end of a light spring is attached to a rigid horizontal support.

An object W of mass 0.15 kg is suspended from the other end of the spring. The extension x of the spring is proportional to the force F causing the extension. The force per unit extension of the spring k is $18 \mathrm{Nm}^{-1}$ .

A student pulls W down such that the extension of the spring increases by 0.040 m . The student releases W and as a result W performs simple harmonic motion (SHM).

State what is meant by the expression "W performs SHM".

[Maximum number: 4]

This question is in two parts.

The graph shows how the velocity v of particle P varies with time t.

Use the graph opposite to determine for the motion of P the

amplitude.

Marking guidance:

max velocity $=0.74( \pm 0.02) \mathrm{ms}^{-1}$ ;
recognize max velocity $=\omega x_{0}$ ;
$\omega=\left(\frac{2 \pi}{T}=\frac{2 \pi}{0.30}=\right) 20.9 \mathrm{rads}^{-1} ;$ $x_{0}=\left(\frac{0.74}{20.9}=\right) 3.5( \pm 0.2) \times 10^{-2} \mathrm{~m} ;$
or
identifies displacement with area; uses one quarter of a cycle;
answer in the range of 30 to 40 mm ;
answer in the range of 33 to 37 mm ;

[Maximum number: 3]

One end of a light spring is attached to a rigid horizontal support.

An object W of mass 0.15 kg is suspended from the other end of the spring. The extension x of the spring is proportional to the force F causing the extension. The force per unit extension of the spring k is $18 \mathrm{Nm}^{-1}$ .

A student pulls W down such that the extension of the spring increases by 0.040 m . The student releases W and as a result W performs simple harmonic motion (SHM).

Determine the period of oscillation of the spring.

$\omega=\sqrt{\frac{a}{x}}$ ;
$=10.95 \mathrm{~rad} \mathrm{~s}^{-2}$ ;
$T=\left(\frac{2\pi}{\omega}=\right) \frac{6.28}{10.95}=0.57 \mathrm{~s}$ ;

[Maximum number: 1]

An object oscillates at the free end of a vertical spring. The graph shows the variation of the object's position with time.

At which position does the object have zero velocity and a negative acceleration?

[Maximum number: 2]

A mass is attached to a vertical spring. The other end of the spring is attached to the driver of an oscillator.

The mass is performing very lightly damped harmonic oscillations. The frequency of the driver is higher than the natural frequency of the system. At one instant the driver is moving downwards.

State and explain the direction of motion of the mass at this instant.

because the mass and the driver are out of phase «by $\pi » \vee$ so upwards

Justification needed for MP2

[Maximum number: 2]

A boat is moved from land to water by rolling it across a set of cylindrical airbags.

During the testing of the rod, it is rotated about its central axis. Point P is on the surface of the rod. At time t=0, P is at a point $45^{\circ}$ above the horizontal. The linear speed of P is $2.0 \mathrm{~ms}^{-1}$ .

The vertical displacement x of P can be modelled with an equation $x=x_{0} \sin (\omega t+\phi)$ .

Determine the vertical velocity of P at $t=3.0 \mathrm{~s}$ .

Use of $v=\omega x_{0} \cos (\omega t+\phi)$ and $\phi=\frac{\pi}{4}$ $v=\omega x_{0} \cos (\omega t+\phi)=(2.22)(0.9) \cos \left((2.22)(3)+\frac{\pi}{4}\right)=0.79 m s^{-1} \quad$

IB Physics HL C 1 Simple Harmonic Motion Questions

Exam points

Covered syllabus points

Covered syllabus points

Question 1

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