Edexcel A Level Further Maths Paper 4C: Further Mechanics 2 October 2020 question paper 9FM0/4C

Three particles of masses 3m, 4m and 2m are placed at the points (−2, 2), (3, 1) and (p, p) respectively.

The value of p is such that the distance of the centre of mass of the three particles from the point (0, 0) is as small as possible.

Find the value of p.

A uniform plane figure R, shown shaded in Figure 1, is bounded by the x-axis, the line with equation x= ln 5, the curve with equation y= 8e^−x and the line with equation x = ln 2. The unit of length on each axis is one metre.

The area of R is 2.4 m²

The centre of mass of R is at the point with coordinates (

Use algebraic integration to show that = 1.4

Figure 2 shows a uniform lamina ABCD, which is the same size and shape as R. The lamina is freely suspended from C and hangs in equilibrium with CB at an angle θ ° to the downward vertical.

Find the value of θ

A particle P of mass 0.5 kg is moving along the positive x-axis in the direction of x increasing. At time t seconds (t 0), P is x metres from the origin O and the speed of P is vm s⁻¹. The resultant force acting on P is directed towards O and has magnitude kv²N, where k is a positive constant.

When x= 1, v= 4 and when x= 2, v= 2

Show that v=ab^x, where a and b are constants to be found.

The time taken for the speed of P to decrease from 4 m s⁻¹ to 2 m s⁻¹ is T seconds.

A uniform solid cylinder of base radius r and height r has the same density as a uniform solid hemisphere of radius r. The plane face of the hemisphere is joined to a plane face of the cylinder to form the composite solid S shown in Figure 3. The point O is the centre of the plane face of S.

Show that the distance from O to the centre of mass of S is

The solid S is placed with its plane face on a rough horizontal plane. The coefficient of friction between S and the plane is μ. A horizontal force P is applied to the highest point of S. The magnitude of P is gradually increased.

Find the range of values of μ for which S will slide before it starts to tilt.

A particle P of mass 0.75 kg is attached to one end of a light inextensible string of length 60 cm. The other end of the string is attached to a fixed point A that is vertically above the point O on a smooth horizontal table, such that OA = 40 cm. The particle remains in contact with the table, with the string taut, and moves in a horizontal circle with centre O, as shown in Figure 4.

The particle is moving with a constant angular speed of 3 radians per second.

(a) Find (i) the tension in the string,

the normal reaction between P and the table.

The angular speed of P is now gradually increased.

Find the angular speed of P at the instant P loses contact with the table.

A particle P of mass m is attached to one end of a light inextensible string of length l. The other end of the string is attached to a fixed point O. The particle is held with the string taut and OP horizontal. The particle is then projected vertically downwards with speed u, where u²= gl. When OP has turned through an angle α and the string is still taut, the speed of P is v, as shown in Figure 5. At this instant the tension in the string is T.

Find, in terms of g and l, the speed of P at the instant when the string goes slack.

Find, in terms of l, the greatest vertical height reached by P above the level of O.

A light elastic spring has natural length l and modulus of elasticity 4mg. A particle P of mass m is attached to one end of the spring. The other end of the spring is attached to a fixed point A. The point B is vertically below A with AB= . The particle P is released from rest at B.

Find, in terms of m, l and g, the maximum kinetic energy of P during the motion.

Find the time within each complete oscillation for which the length of the spring is less than l.