CAIE M1 (Mechanics 1) 2003 June

1 A crate of mass 800 kg is lifted vertically, at constant speed, by the cable of a crane. Find

1. the tension in the cable,
2. the power applied to the crate in increasing the height by 20 m in 50 s .

2
\includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-2_405_384_550_884} Three coplanar forces of magnitudes \(10 \mathrm {~N} , 10 \mathrm {~N}\) and 6 N act at a point \(P\) in the directions shown in the diagram. \(P Q\) is the bisector of the angle between the two forces of magnitude 10 N .

1. Find the component of the resultant of the three forces
   (a) in the direction of \(P Q\),
   (b) in the direction perpendicular to \(P Q\).
2. Find the magnitude of the resultant of the three forces.

3
\includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-2_556_974_1548_587} The diagram shows the velocity-time graphs for the motion of two cyclists \(P\) and \(Q\), who travel in the same direction along a straight path. Both cyclists start from rest at the same point \(O\) and both accelerate at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) up to a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Both then continue at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \(Q\) starts his journey \(T\) seconds after \(P\).

1. Show in a sketch of the diagram the region whose area represents the displacement of \(P\), from \(O\), at the instant when \(Q\) starts. Given that \(P\) has travelled 16 m at the instant when \(Q\) starts, find
2. the value of \(T\),
3. the distance between \(P\) and \(Q\) when \(Q\) 's speed reaches \(10 \mathrm {~ms} ^ { - 1 }\).

4 A particle moves in a straight line. Its displacement \(t\) seconds after leaving the fixed point \(O\) is \(x\) metres, where \(x = \frac { 1 } { 2 } t ^ { 2 } + \frac { 1 } { 30 } t ^ { 3 }\). Find

1. the speed of the particle when \(t = 10\),
2. the value of \(t\) for which the acceleration of the particle is twice its initial acceleration.

5
\includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-3_504_387_598_881}
\(S _ { 1 }\) and \(S _ { 2 }\) are light inextensible strings, and \(A\) and \(B\) are particles each of mass 0.2 kg . Particle \(A\) is suspended from a fixed point \(O\) by the string \(S _ { 1 }\), and particle \(B\) is suspended from \(A\) by the string \(S _ { 2 }\). The particles hang in equilibrium as shown in the diagram.

1. Find the tensions in \(S _ { 1 }\) and \(S _ { 2 }\). The string \(S _ { 1 }\) is cut and the particles fall. The air resistance acting on \(A\) is 0.4 N and the air resistance acting on \(B\) is 0.2 N .
2. Find the acceleration of the particles and the tension in \(S _ { 2 }\).

6 A small block of mass 0.15 kg moves on a horizontal surface. The coefficient of friction between the block and the surface is 0.025 .

1. Find the frictional force acting on the block.
2. Show that the deceleration of the block is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The block is struck from a point \(A\) on the surface and, 4 s later, it hits a boundary board at a point \(B\). The initial speed of the block is \(5.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the distance \(A B\). The block rebounds from the board with a speed of \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves along the line \(B A\). Find
4. the speed with which the block passes through \(A\),
5. the total distance moved by the block, from the instant when it was struck at \(A\) until the instant when it comes to rest.

7
\includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-4_257_988_267_580} The diagram shows a vertical cross-section \(A B C D\) of a surface. The parts \(A B\) and \(C D\) are straight and have lengths 2.5 m and 5.2 m respectively. \(A D\) is horizontal, and \(A B\) is inclined at \(60 ^ { \circ }\) to the horizontal. The points \(B\) and \(C\) are at the same height above \(A D\). The parts of the surface containing \(A B\) and \(B C\) are smooth. A particle \(P\) is given a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\), in the direction \(A B\), and it subsequently reaches \(D\). The particle does not lose contact with the surface during this motion.

1. Find the speed of \(P\) at \(B\).
2. Show that the maximum height of the cross-section, above \(A D\), is less than 3.2 m .
3. State briefly why \(P\) 's speed at \(C\) is the same as its speed at \(B\).
4. The frictional force acting on the particle as it travels from \(C\) to \(D\) is 1.4 N . Given that the mass of \(P\) is 0.4 kg , find the speed with which \(P\) reaches \(D\).