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Date	November 2017	Marks available	2	Reference code	17N.3.SL.TZ0.7
Level	Standard level	Paper	Paper 3	Time zone	0 - no time zone
Command term	Label and Draw	Question number	7	Adapted from	N/A

Question

A hoop of mass m, radius r and moment of inertia mr² rests on a rough plane inclined at an angle θ to the horizontal. It is released so that the hoop gains linear and angular acceleration by rolling, without slipping, down the plane.

On the diagram, draw and label the forces acting on the hoop.

[2]

Show that the linear acceleration a of the hoop is given by the equation shown.

a = $\frac{{g \times \sin q}}{2}$

[4]

Calculate the acceleration of the hoop when θ = 20°. Assume that the hoop continues to roll without slipping.

[1]

State the relationship between the force of friction and the angle of the incline.

[2]

The angle of the incline is slowly increased from zero. Determine the angle, in terms of the coefficient of friction, at which the hoop will begin to slip.

[3]

Markscheme

weight, normal reaction and friction in correct direction

correct points of application for at least two correct forces

Labelled on diagram.

Allow different wording and symbols

Ignore relative lengths

ALTERNATIVE 1

ma = mg sin θ – F_f

I $\alpha$ = F_f x r

mr $\alpha$ = F_f

$\alpha$ = $\frac{a}{r}$

ma = mg sin θ – mr $\frac{a}{r}$ → 2a = g sin θ

Can be in any order

No mark for re-writing given answer

Accept answers using the parallel axis theorem (with I = 2mr²) only if clear and explicit mention that the only torque is from the weight

Answer given look for correct working

ALTERNATIVE 2

mgh = $\frac{1}{2}$ Iω² + $\frac{1}{2}$ mv²

substituting ω = $\frac{v}{r}$ «giving v = $\sqrt {gh}$ »

correct use of a kinematic equation

use of trigonometry to relate displacement and height «s = h sin θ»

For alternative 2, MP3 and MP4 can only be awarded if the previous marking points are present

1.68 «ms^–2»

ALTERNATIVE 1

N = mg cos θ

F_f ≤ μmg cos θ

ALTERNATIVE 2

F_f = ma «from 7(b)»

so F_f = $\frac{{mg\sin \theta }}{2}$

F_f = μmg cos θ

$\frac{{mg\sin \theta }}{2}$ = mg sin θ – μmg cos θ

mg $\frac{{\sin \theta }}{2}$ = μmg cos θ

algebraic manipulation to reach tan θ = 2μ

Examiners report

[N/A]

Syllabus sections

Option B: Engineering physics » Option B: Engineering physics (Core topics) » B.1 – Rigid bodies and rotational dynamics

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18M.3.SL.TZ2.6a.iii:
Determine the angular velocity of the wheel at B.
18M.3.SL.TZ2.6b.ii:
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18N.3.SL.TZ0.6b.ii:
After time t the rod makes an angle θ with the horizontal. Outline why the equation $\theta = \frac{1}{2}\alpha {t^2}$ cannot be used to find the time it takes θ to become $\frac{\pi }{2}$ (that is for the rod to become vertical for the first time).
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18N.3.SL.TZ0.6a: Calculate the force the support exerts on the rod.
18M.3.SL.TZ2.6a.ii:
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18M.3.SL.TZ1.6b.ii:
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17M.3.SL.TZ2.6a.i:
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16N.3.SL.TZ0.8a:
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18N.3.SL.TZ0.6c.i:
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18N.3.SL.TZ0.6b.i:
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17N.3.SL.TZ0.6a: Explain what is meant by proper length.
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19N.3.SL.TZ0.5a: State the torque provided by the force W about the axis of the flywheel.
19N.3.SL.TZ0.5b(iii):
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19N.3.SL.TZ0.5b(i): Identify the physical quantity represented by the area under the graph.
17N.3.SL.TZ0.7d:
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16N.3.SL.TZ0.8b:
Show that the torque acting on the flywheel is about 0.3 Nm.
17M.3.SL.TZ1.5b.ii:
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18M.3.SL.TZ2.6a.i:
The moment of inertia of the wheel is 1.3 × 10^–4 kg m². Outline what is meant by the moment of inertia.
18M.3.SL.TZ1.6c:
Calculate the new angular speed of the rotating system.
19M.3.SL.TZ2.8b:
Calculate F.
18M.3.SL.TZ1.6b.i:
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20N.3.SL.TZ0.6a:
Show that the final angular velocity of the bar is about $3 rad s^{- 1}$ .
16N.3.SL.TZ0.8c:
(i) Calculate the tension in the string.

(ii) Determine the mass m of the falling object.
17N.3.SL.TZ0.7e:
The angle of the incline is slowly increased from zero. Determine the angle, in terms of the coefficient of friction, at which the hoop will begin to slip.
19N.3.SL.TZ0.5c(ii):
At t = 5.00 s the flywheel is spinning with angular velocity 200 rad s^–1. The support bearings exert a constant frictional torque on the axle. The flywheel comes to rest after 8.00 × 10³ revolutions. Calculate the magnitude of the frictional torque exerted on the flywheel.
19M.3.SL.TZ2.9a:
Show that the total kinetic energy E_k of the sphere when it rolls, without slipping, at speed v is ${E_{\text{K}}} = \frac{7}{{10}}m{v^2}$ .
20N.3.SL.TZ0.6d:
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20N.3.SL.TZ0.7a:
Explain the direction in which the person-turntable system starts to rotate.
20N.3.SL.TZ0.6c:
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20N.3.SL.TZ0.8:
A solid sphere of radius $r$ and mass $m$ is released from rest and rolls down a slope, without slipping. The vertical height of the slope is $h$ . The moment of inertia $I$ of this sphere about an axis through its centre is $\frac{2}{5} m r^{2}$ .

Show that the linear velocity $v$ of the sphere as it leaves the slope is $\sqrt{\frac{10 g h}{7}}$ .
17M.3.SL.TZ2.6b:
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17N.3.SL.TZ0.7b:
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Determine in terms of M and v the energy lost during the collision.
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19M.3.SL.TZ2.9b:
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Calculate the speed of the sphere at the bottom of the ramp.

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Option B: Engineering physics » Option B: Engineering physics (Core topics)

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