Standard +0.3 This is a straightforward application of rotational dynamics formulas (moment of inertia for a disc, torque, angular deceleration) with clear given values and standard two-part calculation. While it's Further Maths content, the problem requires only direct substitution into standard formulas with no conceptual challenges or novel problem-solving, making it slightly easier than average overall.
A circular wheel is modelled as a uniform disc of mass \(6\) kg and radius \(0.25\) m. It is rotating with angular speed \(2\) rad s\(^{-1}\) about a fixed smooth axis perpendicular to its plane and passing through its centre. A braking force of constant magnitude is applied tangentially to the rim of the wheel. The wheel comes to rest \(5\) s after the braking force is applied. Find the magnitude of the braking force and the angle turned through by the wheel while the braking force acts. [7]
Find MI of wheel: I = ½×6×0⋅25 2 [= 3/16 = 0⋅1875] B1
Find angular deceln. from ω, t (ignoring sign):
d 2θ /dt 2
= ω / t = 2/5 or 0⋅4 M1 A1
Use moment eqn to find magnitude of braking force: F = I
d 2θ /dt 2
/ r = 3/10 or 0⋅3 [N] M1 A1
Find angle turned by wheel: θ = ω 2 /2
d 2θ /dt 2
[= 2 2 / (2×0⋅4)]
or ½ωt [= ½ × 2 × 5]
or ωt + ½ d 2θ /dt 2 t 2 [= 10 – 5]
or ½ I (dθ/dt) 2 / r F
[= ½ (3/16) 2 2 / (0⋅25 × 0⋅3)] M1
Answer
Marks
= 5 [rad] A1
(5)
(2)
[7]
Question 2:
2 | Find MI of wheel: I = ½×6×0⋅25 2 [= 3/16 = 0⋅1875] B1
Find angular deceln. from ω, t (ignoring sign): |d 2θ /dt 2| = ω / t = 2/5 or 0⋅4 M1 A1
Use moment eqn to find magnitude of braking force: F = I |d 2θ /dt 2| / r = 3/10 or 0⋅3 [N] M1 A1
Find angle turned by wheel: θ = ω 2 /2 |d 2θ /dt 2| [= 2 2 / (2×0⋅4)]
or ½ωt [= ½ × 2 × 5]
or ωt + ½ d 2θ /dt 2 t 2 [= 10 – 5]
or ½ I (dθ/dt) 2 / r F
[= ½ (3/16) 2 2 / (0⋅25 × 0⋅3)] M1
= 5 [rad] A1 | (5)
(2) | [7]
A circular wheel is modelled as a uniform disc of mass $6$ kg and radius $0.25$ m. It is rotating with angular speed $2$ rad s$^{-1}$ about a fixed smooth axis perpendicular to its plane and passing through its centre. A braking force of constant magnitude is applied tangentially to the rim of the wheel. The wheel comes to rest $5$ s after the braking force is applied. Find the magnitude of the braking force and the angle turned through by the wheel while the braking force acts. [7]
\hfill \mbox{\textit{CAIE FP2 2009 Q2 [7]}}