2 A cyclist and her bicycle have a combined mass of 80 kg .
- Initially, the cyclist accelerates from rest to \(3 \mathrm {~ms} ^ { - 1 }\) against negligible resistances along a horizontal road.
(A) How much energy is gained by the cyclist and bicycle?
(B) The cyclist travels 12 m during this acceleration. What is the average driving force on the bicycle? - While exerting no driving force, the cyclist free-wheels down a hill. Her speed increases from \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). During this motion, the total work done against friction is 1600 J and the drop in vertical height is \(h \mathrm {~m}\).
Without assuming that the hill is uniform in either its angle or roughness, calculate \(h\).
- The cyclist reaches another horizontal stretch of road and there is now a constant resistance to motion of 40 N .
(A) When the power of the driving force on the bicycle is a constant 200 W , what constant speed can the cyclist maintain?
(B) Find the power of the driving force on the bicycle when travelling at a speed of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with an acceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b2962c91-4739-4d1e-98f3-62d420f6dddf-4_671_760_296_310}
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\caption{Fig. 3.1}
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\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b2962c91-4739-4d1e-98f3-62d420f6dddf-4_703_622_264_1213}
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\caption{Fig. 3.2}
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A lamina is made from uniform material in the shape shown in Fig.3.1. BCJA, DZOJ, ZEIO and FGHI are all rectangles. The lengths of the sides are shown in centimetres. - Find the coordinates of the centre of mass of the lamina, referred to the axes shown in Fig. 3.1.
The rectangles BCJA and FGHI are folded through \(90 ^ { \circ }\) about the lines CJ and FI respectively to give the fire-screen shown in Fig. 3.2.
- Show that the coordinates of the centre of mass of the fire-screen, referred to the axes shown in Fig. 3.2, are (2.5, 0, 57.5).
The \(x\) - and \(y\)-axes are in a horizontal floor. The fire-screen has a weight of 72 N . A horizontal force \(P \mathrm {~N}\) is applied to the fire-screen at the point Z . This force is perpendicular to the line DE in the positive \(x\) direction. The fire-screen is on the point of tipping about the line AH .
- Calculate the value of \(P\).
The coefficient of friction between the fire-screen and the floor is \(\mu\).
- For what values of \(\mu\) does the fire-screen slide before it tips?