IB Physics HL C.1 Simple Harmonic Motion Question Bank

Practise IB Physics HL C.1 by analysing oscillator equations, phase, energy and data-driven spring or pendulum motion at HL depth.

Exam points

Solve SHM equations for displacement, velocity and acceleration while interpreting phase and initial conditions.
Connect angular frequency, period and restoring force to spring and pendulum models.
Evaluate kinetic, potential and total energy through graphs and extended oscillation evidence.

Covered syllabus points

C.1.1—Conditions for SHM C.1.2—SHM defining equation C.1.3—SHM quantities C.1.4—Period, frequency and angular frequency C.1.5—Energy in SHM C.1.6—SHM graphs C.1.7—SHM from circular motion C.1.8 (HL)—Phase angle C.1.9 (HL)—SHM equations

[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).

Figure for Question C.1 Simple harmonic motion question 2 — IB Physics HL

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 $x=x_{0} \cos \omega t$ ;
$x\left(=35 \cos \left[\frac{2 \pi}{0.3} \times 0.2\right]\right)=17.5 \mathrm{~mm} ;$

All question bank results loaded

Covered syllabus points

C.1 Simple harmonic motion question 1

Question (a)

Question (i)

Question (ii)

C.1 Simple harmonic motion question 2

Question (a)

Question (b)

Question (i)

Question (ii)

Question (iii)

Question (c)

Question (i)

Question (ii)

Question (d)

C.1 Simple harmonic motion question 3

Question (a)

Question (i)

Question (b)

Question (i)

Question (ii)

Question (iii)