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Date	November 2020	Marks available	3	Reference code	20N.3.SL.TZ0.8
Level	Standard level	Paper	Paper 3	Time zone	0 - no time zone
Command term	Show that	Question number	8	Adapted from	N/A

Question

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 $\sqrt{\frac{10 g h}{7}}$ .

Markscheme

conservation of rotational and linear energy

OR
$m g h = \frac{1}{2} m v^{2} + \frac{1}{2} I ω^{2}$ ✓

using $I = \frac{2}{5} m r^{2}$ AND $ω = \frac{v}{r}$ ✓

with correct manipulation to find the requested relationship ✓

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).
18N.3.SL.TZ0.6c.ii: At the instant the rod becomes vertical calculate the angular momentum of the rod.
18N.3.SL.TZ0.6a: Calculate the force the support exerts on the rod.
18M.3.SL.TZ2.6a.ii:
In moving from point A to point B, the centre of mass of the wheel falls through a vertical distance of 0.36 m. Show that the translational speed of the wheel is about 1 m s^–1 after its displacement.
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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:
At the instant the rod becomes vertical show that the angular speed is ω = 2.43 rad s^–1.
18N.3.SL.TZ0.6b.i:
Calculate, in rad s^–2, the initial angular acceleration $\alpha$ of the rod.
17N.3.SL.TZ0.6a: Explain what is meant by proper length.
17N.3.SL.TZ0.7a: On the diagram, draw and label the forces acting on the hoop.
19N.3.SL.TZ0.5b(ii): Show that the angular velocity of the flywheel at t = 5.00 s is 200 rad s–1.
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):
Calculate the maximum tension in the string.
19N.3.SL.TZ0.5c(i): The flywheel is in translational equilibrium. Distinguish between translational equilibrium...
19N.3.SL.TZ0.5b(i): Identify the physical quantity represented by the area under the graph.
17N.3.SL.TZ0.7d:
State the relationship between the force of friction and the angle of the incline.
16N.3.SL.TZ0.8b:
Show that the torque acting on the flywheel is about 0.3 Nm.
17M.3.SL.TZ1.5b.ii:
Calculate the number of revolutions made by the system before it comes to rest.
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:
Calculate, for the merry-go-round after one revolution, the angular speed.
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.TZ0.6b:
Draw the variation with time $t$ of the angular displacement $θ$ of the bar during the acceleration.
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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:
Calculate the torque acting on the bar while it is accelerating.
17M.3.SL.TZ2.6b:
The forces act for 2.00 s. Show that the final angular speed of the probe is about 16 rad $\,$ s^–1.
17N.3.SL.TZ0.7b:
Show that the linear acceleration a of the hoop is given by the equation shown.

a = $\frac{{g \times \sin q}}{2}$
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17M.3.SL.TZ1.5a.i:
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17M.3.SL.TZ1.5a.ii:
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17M.3.SL.TZ1.5a.iii:
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17M.3.SL.TZ1.5a.iv:
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18M.3.SL.TZ1.6a:
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19M.3.SL.TZ2.9b:
A solid sphere of mass 1.5 kg is rolling, without slipping, on a horizontal surface with a speed of 0.50 m s^-1. The sphere then rolls, without slipping, down a ramp to reach a horizontal surface that is 45 cm lower.

Calculate the speed of the sphere at the bottom of the ramp.

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Option B: Engineering physics