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Question 23M.3.HL.TZ2.8

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Date May 2023 Marks available [Maximum mark: 10] Reference code 23M.3.HL.TZ2.8
Level HL Paper 3 Time zone TZ2
Command term Calculate, Determine, Outline, Show that Question number 8 Adapted from N/A
8.
[Maximum mark: 10]
23M.3.HL.TZ2.8

A student models a rotating dancer using a system that consists of a vertical cylinder, a horizontal rod and two spheres.

The cylinder rotates from rest about the central vertical axis. A rod passes through the cylinder with a sphere on each side of the cylinder. Each sphere can move along the rod. Initially the spheres are close to the cylinder.

A horizontal force of 50 N is applied perpendicular to the rod at a distance of 0.50 m from the central axis. Another horizontal force of 40 N is applied in the opposite direction at a distance of 0.20 m from the central axis. Air resistance is negligible.

(a)

Show that the net torque on the system about the central axis is approximately 30 N m.

[1]

Markscheme

ΣΓ = 50 × 0.5 + 40 × 0.2

OR

33 «Nm» ✓

 

Accept opposite rotational sign convention

(b)

The system rotates from rest and reaches a maximum angular speed of 20 rad s−1 in a time of 5.0 s. Calculate the angular acceleration of the system.

[1]

Markscheme

«α = 205 =» 4 «rad s−2» ✓

(c)

Determine the moment of inertia of the system about the central axis.

[2]

Markscheme

I=Γα

OR

33 = I × 4 ✓

 

I = 8.25 «kg m2» ✓

 

Allow ECF from (a) and (b)

Award [2] for a BCA

(d)

When the system has reached its maximum angular speed, the two forces are removed. The spheres now move outward, away from the central axis.

(d.i)

Outline why the angular speed ω decreases when the spheres move outward.

[2]

Markscheme

moment of inertia increases ✓

Angular momentum is conserved ✓


Allow algebraic expressions e.g. ω = LI so ω decreases for MP2

(d.ii)

Show that the rotational kinetic energy is 12Lω where L is the angular momentum of the system.

[1]

Markscheme

Ek «= 12Iω2 =» 12(Iω)ω = 12Lω ✓

 

Accept equivalent methods

(d.iii)

When the spheres move outward, the angular speed decreases from 20 rad s−1 to 12 rad s−1. Calculate the percentage change in rotational kinetic energy that occurs when the spheres move outward.

[2]

Markscheme

«Ek =» 12Lω1 = 12Lω2 ✓

OR

Ek1Ek2=ω1ω2

OR

«L is constant so» Ek is proportional to ω ✓

40 % «energy loss» ✓

 

MP1 is for understanding that angular momentum is constant so change in rotational kinetic energy is proportional to change in angular velocity

Award [0] if E = 0.5 I ω2 is used with the same I value for both values of E

Award [2] for BCA

(e)

Outline one reason why this model of a dancer is unrealistic.

[1]

Markscheme

one example specified eg friction, air resistance, mass distribution not modelled ✓

 

Award [1] for any reasonable physical parameter that is not consistent with the model

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