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*N35408A0228*

1. A cyclist starts from rest and moves along a straight horizontal road. The combined mass

of the cyclist and his cycle is 120 kg. The resistance to motion is modelled as a constant

force of magnitude 32 N. The rate at which the cyclist works is 384 W. The cyclist

accelerates until he reaches a constant speed ofv m s1.

Find

(a) the value ofv,

(3)

(b) the acceleration of the cyclist at the instant when the speed is 9 m s1.

(3)

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*N35408A0328* Turn over

Question 1 continued

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(Total 6 marks)

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*N35408A0428*

2. A particle of mass 2 kg is moving with velocity (5i+j) m s1 when it receives an impulse

of (6i+ 8j) N s. Find the kinetic energy of the particle immediately after receiving the

impulse.

(5)

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*N35408A0528* Turn over

Question 2 continued

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(Total 5 marks)

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*N35408A0628*

3. A particle moves along thex-axis. At time t= 0the particle passes through the origin with

speed 8 m s1 in the positivex-direction. The acceleration of the particle at time tseconds,

t0, is (4t3 12t) m s2 in the positivex-direction.

Find

(a) the velocity of the particle at time tseconds,(3)

(b) the displacement of the particle from the origin at time tseconds,

(2)

(c) the values oftat which the particle is instantaneously at rest.

(3)

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*N35408A0728* Turn over

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8

*N35408A0828*

Question 3 continued

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*N35408A0928* Turn over

Question 3 continued

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(Total 8 marks)

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*N35408A01028*

4.

Figure 1

A box of mass 30 kg is held at rest at point A on a rough inclined plane. The plane is

inclined at 20 to the horizontal. PointB is 50 m fromA up a line of greatest slope of the

plane, as shown in Figure 1. The box is dragged fromA toB by a force acting parallel to

AB and then held at rest at B. The coefficient of friction between the box and the plane

is1

4

. Friction is the only non-gravitational resistive force acting on the box. Modelling

the box as a particle,

(a) find the work done in dragging the box fromA toB.

(6)

The box is released from rest at the pointB and slides down the slope. Using the work-

energy principle, or otherwise,

(b) find the speed of the box as it reachesA.

(5)

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A

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*N35408A01128* Turn over

Question 4 continued

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12

*N35408A01228*

Question 4 continued

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*N35408A01328* Turn over

Question 4 continued

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(Total 11 marks)

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*N35408A01428*

5.

Figure 2

The uniform L-shaped laminaABCDEF, shown in Figure 2, has sidesAB andFEparallel,

and sidesBCandED parallel. The pairs of parallel sides are 9 cm apart. The pointsA,F,

D and Clie on a straight line.

AB=BC= 36 cm,FE=ED= 18 cm. ABC=FED= 90, and

BCD=EDF=EFD =BAC=45.

(a) Find the distance of the centre of mass of the lamina from

(i) sideAB,

(ii) sideBC.

(7)

The lamina is freely suspended fromA and hangs in equilibrium.

(b) Find, to the nearest degree, the size of the angle betweenAB and the vertical.

(3)

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A

B C

DE

F

9 cm

18 cm36 cm

9 cm

36 cm

18 cm

45

45

45

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*N35408A01528* Turn over

Question 5 continued

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*N35408A01628*

Question 5 continued

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*N35408A01728* Turn over

Question 5 continued

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(Total 10 marks)

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*N35408A01828*

6. [In this question, the unit vectors i andj are in a vertical plane, i being horizontal andj

being vertically upwards.]

Figure 3

At time t = 0, a particle P is projected from the point A which has position vector

10j metres with respect to a fixed origin O at ground level. The ground is horizontal. The

velocity of projection ofP is (3i+ 5j) m s1, as shown in Figure 3. The particle moves

freely under gravity and reaches the ground afterTseconds.

(a) For 0tT, show that, with respect to O, the position vector, r metres, ofPat time

tseconds is given by

r =3ti+(10+5t4.9t2)j

(3)

(b) Find the value ofT.

(3)

(c) Find the velocity ofPat time tseconds (0tT).

(2)

WhenPis at the pointB, the direction of motion ofPis 45below the horizontal.

(d) Find the time taken forPto move fromA toB.

(2)

(e) Find the speed ofPas it passes throughB.

(2)

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10 m

1(3 5 ) m s+i j

A

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*N35408A01928* Turn over

Question 6 continued

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*N35408A02028*

Question 6 continued

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*N35408A02128* Turn over

Question 6 continued

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(Total 12 marks)

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*N35408A02228*

7.

Figure 4

A uniform plankAB, of weight 100 N and length 4 m, rests in equilibrium with the endA

on rough horizontal ground. The plank rests on a smooth cylindrical drum. The drum is

fixed to the ground and cannot move. The point of contact between the plank and the drum

is C, whereAC= 3 m, as shown in Figure 4. The plank is resting in a vertical plane whichis perpendicular to the axis of the drum, at an angle to the horizontal, where sin =

1

3.

The coefficient of friction between the plank and the ground is . Modelling the plank as

a rod, find the least possible value of.

(10)

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4 m

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*N35408A02328* Turn over

Question 7 continued

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*N35408A02428*

Question 7 continued

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*N35408A02528* Turn over

Question 7 continued

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(Total 10 marks)

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*N35408A02628*

8. A particle Pof mass m kg is moving with speed 6 m s1 in a straight line on a smooth

horizontal floor. The particle strikes a fixed smooth vertical wall at right angles and

rebounds. The kinetic energy lost in the impact is 64 J. The coefficient of restitution

betweenPand the wall is13

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(a) Show that m = 4. (6)

After rebounding from the wall, Pcollides directly with a particle Q which is moving

towards Pwith speed 3 m s1. The mass ofQ is 2 kg and the coefficient of restitution

betweenPand Q is1

3.

(b) Show that there will be a second collision betweenPand the wall.

(7)

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*N35408A02728* Turn over

Question 8 continued

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Question 8 continued

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TOTAL FOR PAPER: 75 MARKS

Q8

(Total 13 marks)