OCR M2 (Mechanics 2) 2011 January

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\includegraphics[max width=\textwidth, alt={}, center]{941c0c81-a74f-49c0-acb7-1c23266fc2c8-02_378_471_260_836} A uniform square frame \(A B C D\) has sides of length 0.6 m . The side \(A D\) is removed from the frame, and the open frame \(A B C D\) is attached at \(A\) to a fixed point (see diagram).

1. Calculate the distance of the centre of mass of the open frame from \(A\). The open frame rotates about \(A\) in the plane \(A B C D\) with angular speed \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
2. Calculate the speed of the centre of mass of the open frame.

2 The resistance to the motion of a car is \(k v ^ { \frac { 3 } { 2 } } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed and \(k\) is a constant. The power exerted by the car's engine is 15000 W , and the car has constant speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a horizontal road.

1. Show that \(k = 4.8\). With the engine operating at a much lower power, the car descends a hill of inclination \(\alpha\), where \(\sin \alpha = \frac { 1 } { 15 }\). At an instant when the speed of the car is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Given that the mass of the car is 700 kg , calculate the power of the engine.
   \includegraphics[max width=\textwidth, alt={}, center]{941c0c81-a74f-49c0-acb7-1c23266fc2c8-02_579_447_1658_849} A particle \(P\) of mass 0.4 kg is attached to one end of each of two light inextensible strings which are both taut. The other end of the longer string is attached to a fixed point \(A\), and the other end of the shorter string is attached to a fixed point \(B\), which is vertically below \(A\). The string \(A P\) makes an angle of \(30 ^ { \circ }\) with the vertical and is 0.5 m long. The string \(B P\) makes an angle of \(60 ^ { \circ }\) with the vertical. \(P\) moves with constant angular speed in a horizontal circle with centre vertically below \(B\) (see diagram). The tension in the string \(A P\) is twice the tension in the string \(B P\). Calculate

4 A block of mass 25 kg is dragged 30 m up a slope inclined at \(5 ^ { \circ }\) to the horizontal by a rope inclined at \(20 ^ { \circ }\) to the slope. The tension in the rope is 100 N and the resistance to the motion of the block is 70 N . The block is initially at rest. Calculate

1. the work done by the tension in the rope,
2. the change in the potential energy of the block,
3. the speed of the block after it has moved 30 m up the slope.

5 A uniform solid is made of a hemisphere with centre \(O\) and radius 0.6 m , and a cylinder of radius 0.6 m and height 0.6 m . The plane face of the hemisphere and a plane face of the cylinder coincide. (The formula for the volume of a sphere is \(\frac { 4 } { 3 } \pi r ^ { 3 }\).)

1. Show that the distance of the centre of mass of the solid from \(O\) is 0.09 m .
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{941c0c81-a74f-49c0-acb7-1c23266fc2c8-03_636_1036_982_593} The solid is placed with the curved surface of the hemisphere on a rough horizontal surface and the axis inclined at \(45 ^ { \circ }\) to the horizontal. The equilibrium of the solid is maintained by a horizontal force of 2 N applied to the highest point on the circumference of its plane face (see diagram). Calculate
   (a) the mass of the solid,
   (b) the set of possible values of the coefficient of friction between the surface and the solid.

6 A small ball \(B\) is projected with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(30 ^ { \circ }\) from a point \(O\) on a horizontal plane, and moves freely under gravity.

1. Calculate the height of \(B\) above the plane when moving horizontally.
   \(B\) has mass 0.4 kg . At the instant when \(B\) is moving horizontally it receives an impulse of magnitude \(I \mathrm { Ns }\) in its direction of motion which immediately increases the speed of \(B\) to \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Calculate \(I\). For the instant when \(B\) returns to the plane, calculate
3. the speed and direction of motion of \(B\),
4. the time of flight, and the distance of \(B\) from \(O\).

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\section*{OCR
RECOGNISING ACHIEVEMENT}