Edexcel A Level Further Maths
Topic questions Mechanics Stability & equilibrium positions
12 questions · 167 marks · about 3.5 hours
The rod makes an angle @ with the upward vertical through 4 and the elastic string is
A uniform rod AB has mass m and length 4a. The end A of the rod is freely hinged to a fixed point. One end of a light elastic string, of natural length a and modulus 
mg, is attached to the end B of the rod. The other end of the string is attached to a small light smooth ring R. The ring can move freely on a smooth horizontal wire which is fixed at a height a above A, and in a vertical plane through A. The angle between the rod and the horizontal is 
, where 0 
θ 
, as shown in Figure 1. Given that the elastic string is vertical,
show that the potential energy of the system is
Show that when θ 
the rod is in stable equilibrium.
Figure 2 shows four uniform rods, each of mass m and length 2 a. The rods are freely hinged at their ends to form a rhombus ABCD. Point A is attached to a fixed point on a ceiling and the rhombus hangs freely with C vertically below A. A light elastic spring of natural length 2 a and modulus of elasticity 7 mg connects the points A and C. A particle of mass 3 m is attached to point C.
Show that, when AD is at an angle 
to the downward vertical, the potential energy V of the system is given by V = 28mga cos2 
– 48mga cosθ + constant
Given that θ > 0
find the value of θ for which the system is in equilibrium,
determine the stability of this position of equilibrium.
A smooth wire, with ends A and B, is in the shape of a semicircle of radius r. The line AB is horizontal and the midpoint of AB is O. The wire is fixed in a vertical plane. A small ring R of mass 2m is threaded on the wire and is attached to two light inextensible strings. One string passes through a small smooth ring fixed at A and is attached to a
Show that the potential energy of the system is
Find the values of 
for which the system is in equilibrium.
Determine the stability of the position of equilibrium for which θ > 0
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A uniform rod AB, of length 2 l and mass 12 m, has its end A smoothly hinged to a fixed point. One end of a light inextensible string is attached to the other end B of the rod. The string passes over a small smooth pulley which is fixed at the point C, where AC is horizontal and AC = 2 l. A particle of mass m is attached to the other end of the string and the particle hangs vertically below C. The angle BAC is 
, where 0 
θ 
as shown in Figure 1.
Show that the potential energy of the system is 4mgl sin 
3sin θ 
constant
Find the value of θ when the system is in equilibrium and determine the stability of this equilibrium position.
A bead B of mass m is threaded on a smooth circular wire of radius r, which is fixed in a vertical plane. The centre of the circle is O, and the highest point of the circle is A. A light elastic string of natural length r and modulus of elasticity kmg has one end attached to the bead and the other end attached to A. The angle between the string and the downward vertical is 
, and the extension in the string is x, as shown in Figure 2. Given that the string is taut,
show that the potential energy of the system is
Given also that k = 3,
find the positions of equilibrium and determine their stability.
___________________________________________________________________________
A small smooth peg P is fixed at a distance d from a fixed smooth vertical wire. A particle of mass 3 m is attached to one end of a light inextensible string which passes over P. The particle hangs vertically below P. The other end of the string is attached to a small ring R of mass m, which is threaded on the wire, as shown in Figure 3.
Show that when R is at a distance x below the level of P the potential energy of the system is
Hence find x, in terms of d, when the system is in equilibrium.
Determine the stability of the position of equilibrium.
A uniform rod AB has mass 4 m and length 4 l. The rod can turn freely in a vertical plane about a fixed smooth horizontal axis through A. A particle of mass km, where k < 7, is attached to the rod at B. One end of a light elastic string, of natural length l and modulus of elasticity 4 mg, is attached to the point D of the rod, where AD = 3 l. The other end of the string is attached to a fixed point E which is vertically above A, where AE = 3 l, as shown in Figure 2. The angle between the rod and the upward vertical is 2
, where
Show that, while the string is stretched, the potential energy of the system is
There is a position of equilibrium with 
Given that k = 4,
show that this position of equilibrium is stable.
Figure 3 shows a framework ABC, consisting of two uniform rods rigidly joined together at B so that 
ABC 
90
. The rod AB has length 2a and mass 4m, and the rod BC has length a and mass 2 m. The framework is smoothly hinged at A to a fixed point, so that the framework can rotate in a fixed vertical plane. One end of a light elastic string, of natural length 2 a and modulus of elasticity 3 mg, is attached to A. The string passes through a small smooth ring R fixed at a distance 2 a from A, on the same horizontal level as A and in the same vertical plane as the framework. The other end of the string is attached to B. The angle ARB is 
, where 0 < 
< 
Show that the potential energy V of the system is given by V = 8amg sin 2
+ 5amg cos 2
+ constant
Find the value of for which the system is in equilibrium.
Determine the stability of this position of equilibrium.
The end A of a uniform rod AB, of length 2 a and mass 4 m, is smoothly hinged to a fixed point. The end B is attached to one end of a light inextensible string which passes over a small smooth pulley, fixed at the same level as A. The distance from A to the pulley is 4 a. The other end of the string carries a particle of mass m which hangs freely, vertically below the pulley, with the string taut. The angle between the rod and the downward vertical is, where, as shown in Figure 1.
Show that the potential energy of the system is
Hence, or otherwise, show that any value of which corresponds to a position of equilibrium of the system satisfies the equation
Given that corresponds to a position of equilibrium, determine its stability.
A light inextensible string of length 2 a has one end attached to a fixed point A. The other end of the string is attached to a particle P of mass m. A second light inextensible string of length L, where L > 
, has one of its ends attached to P and passes over a small smooth peg fixed at a point B. The line AB is horizontal and AB = 2a. The other end of the second string is attached to a particle of mass 
m, which hangs vertically below B, as shown in Figure 2.
Show that the potential energy of the system, when the angle PAB = 2 θ, is 
mga(7sin
10sin 2
) + constant.
Show that there is only one value of cos θ for which the system is in equilibrium and find this value.
Determine the stability of the position of equilibrium.
A uniform rod AB, of length 2 a and mass kM where k is a constant, is free to rotate in a vertical plane about the fixed point A. One end of a light inextensible string of length 6 a is attached to the end B of the rod and passes over a small smooth pulley which is fixed at the point P. The line AP is horizontal and of length 2 a. The other end of the string is attached to a particle of mass M which hangs vertically below the point P, as shown in
Show that the potential energy of the system is
Find the value of k.
Determine the stability of each of the two positions of equilibrium.
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