Equilibrium of rigid bodies

[The centre of mass of a semicircular lamina of radius r is from the centre.]

Figure 2 shows the uniform lamina ABCDE, such that ABDE is a square with sides of length 2 a and BCD is a semicircle with diameter BD.

Show that the distance of the centre of mass of the lamina from BD is

The lamina is freely suspended from D and hangs in equilibrium.

Find, to the nearest degree, the angle that DE makes with the downward vertical.

A uniform rod AB, of mass m and length 2 a, rests with its end A on rough horizontal ground. The rod is held in limiting equilibrium at an angle θ to the horizontal by a light string attached to the rod at B, as shown in Figure 3. The string is perpendicular to the rod and lies in the same vertical plane as the rod. The coefficient of friction between the ground and the rod is

The uniform rectangular lamina ABDE, shown in Figure 1, has side AB of length 2 a and side BD of length 6 a. The point C divides BD in the ratio 1:2 and the point F divides EA in the ratio 1:2. The rectangular lamina is folded along FC to produce the folded lamina L, shown in Figure 2.

Show that the centre of mass of L is a from EF.

The folded lamina, L, is freely suspended from C and hangs in equilibrium.

Find the size of the angle between CF and the downward vertical.

A uniform rod AB, of mass 5 kg and length 8 m, has its end B resting on rough horizontal ground. The rod is held in limiting equilibrium at an angle to the horizontal, where tan = , by a rope attached to the rod at C. The distance AC = 1 m. The rope is in the same vertical plane as the rod. The angle between the rope and the rod is and the tension in the rope is T newtons, as shown in Figure 3. The coefficient of friction between the rod and the ground is . The vertical component of the force exerted on the rod at B by the ground is R newtons.

Find the value of R.

Find the size of angle .

The uniform lamina OBC is one quarter of a circular disc with centre O and radius 4m. The points A and D, on OB and OC respectively, are 3m from O. The uniform lamina ABCD, shown shaded in Figure 1, is formed by removing the triangle OAD from OBC. Given that the centre of mass of one quarter of a uniform circular disc of radius r is at a

find the distance of the centre of mass of the lamina ABCD from AD.

The lamina is freely suspended from D and hangs in equilibrium.

Find, to the nearest degree, the angle between DC and the downward vertical.

A non-uniform rod AB, of mass 5kg and length 4m, rests with one end A on rough horizontal ground. The centre of mass of the rod is d metres from A. The rod is held in limiting equilibrium at an angle to the horizontal by a force P, which acts in a direction perpendicular to the rod at B, as shown in Figure 2. The line of action of P lies in the same vertical plane as the rod.

Find, in terms of d, g and ,

the magnitude of the vertical component of the force exerted on the rod by the ground,

the magnitude of the friction force acting on the rod at A.

Given that tan θ and that the coefficient of friction between the rod and the ground

find the value of d.

A non-uniform rod, AB, of mass m and length 2 l, rests in equilibrium with one end A on a rough horizontal floor and the other end B against a rough vertical wall. The rod is in a vertical plane perpendicular to the wall and makes an angle of 60° with the floor as shown in Figure 1. The coefficient of friction between the rod and the floor is and the coefficient of friction between the rod and the wall is . The rod is on the point of slipping at both ends.

Find the magnitude of the vertical component of the force exerted on the rod by the floor.

The centre of mass of the rod is at G.

Find the distance AG.

The uniform lamina ABCDEF, shown shaded in Figure 1, is symmetrical about the line through B and E. It is formed by removing the isosceles triangle FED, of height 6 a and base 8 a, from the isosceles triangle ABC of height 9 a and base 12 a.

Find, in terms of a, the distance of the centre of mass of the lamina from AC.

The lamina is freely suspended from A and hangs in equilibrium.

Find, to the nearest degree, the size of the angle between AB and the downward vertical.

A uniform rod AB of weight W has its end A freely hinged to a point on a fixed vertical wall. The rod is held in equilibrium, at angle to the horizontal, by a force of magnitude P. The force acts perpendicular to the rod at B and in the same vertical plane as the rod, as shown in Figure 3. The rod is in a vertical plane perpendicular to the wall. The magnitude of the vertical component of the force exerted on the rod by the wall at A is Y.

find the magnitude of the force exerted on the rod by the wall at A, giving your answer in terms of W.

A rough circular cylinder of radius 4 a is fixed to a rough horizontal plane with its axis horizontal. A uniform rod AB, of weight W and length 6aA on the plane and a point C of the rod against the cylinder. The vertical plane through the rod is perpendicular to the axis of the cylinder. The rod is inclined at 60° to the horizontal, as shown in Figure 1.

Show that AC = 4a

The coefficient of friction between the rod and the cylinder is and the coefficient of friction between the rod and the plane is . Given that friction is limiting at both A and C,

The uniform lamina ABCDEF is a regular hexagon with centre O and sides of length 2 m, as shown in Figure 1.

The triangles OAF and OEF are removed to form the uniform lamina OABCDE, shown in Figure 2.

Find the distance of the centre of mass of OABCDE from O.

The lamina OABCDE is freely suspended from E and hangs in equilibrium.

Find the size of the angle between EO and the downward vertical.

A uniform rod AB, of mass m and length 2 a, is freely hinged to a fixed point A. A particle of mass m is attached to the rod at B. The rod is held in equilibrium at an angle to the horizontal by a force of magnitude F acting at the point C on the rod, where AC = b, as shown in Figure 3. The force at C acts at right angles to AB and in the vertical plane containing AB.

Find, in terms of a, b, g, m and ,

the horizontal component of the force acting on the rod at A,

the vertical component of the force acting on the rod at A.

Given that the force acting on the rod at A acts along the rod,

A uniform triangular lamina ABC of mass M is such that AB = AC, BC = 2a and the distance of A from BC is h. A line, parallel to BC and at a distance from A, cuts AB at D and cuts AC at E, as shown in Figure 2. It is given that the mass of the trapezium BCED is

Show that the centre of mass of the trapezium BCED is from BC.

The portion ADE of the lamina is folded through 180° about DE to form the folded lamina shown in Figure 3.

Find the distance of the centre of mass of the folded lamina from BC.

The folded lamina is freely suspended from D and hangs in equilibrium. The angle between DE and the downward vertical is .

Find tan in terms of a and h.

Two uniform rods AB and BC are rigidly joined at B so that ABC = 90°. Rod AB has length 0.5 m and mass 2 kg. Rod BC has length 2 m and mass 3 kg. The centre of mass of the framework of the two rods is at G.

Find the distance of G from BC.

The distance of G from AB is 0.6 m. The framework is suspended from A and hangs freely in equilibrium.

Find the angle between AB and the downward vertical at A.

A ladder, of length 5 m and mass 18 kg, has one end A resting on rough horizontal ground and its other end B resting against a smooth vertical wall. The ladder lies in a vertical plane perpendicular to the wall and makes an angle with the horizontal ground, where tan α = , as shown in Figure 1. The coefficient of friction between the ladder and the ground is μ. A woman of mass 60 kg stands on the ladder at the point C, where AC = 3 m. The ladder is on the point of slipping. The ladder is modelled as a uniform rod and the woman as a particle. Find the value of μ.

Figure 1

A uniform rod AB, of mass 5 kg and length 4 m, has its end A smoothly hinged at a fixed point. The rod is held in equilibrium at an angle of 25° above the horizontal by a force of magnitude F newtons applied to its end B. The force acts in the vertical plane containing the rod and in a direction which makes an angle of 40° with the rod, as shown in Figure 1.

Find the value of F.

Find the magnitude and direction of the vertical component of the force acting on the rod at A.

Figure 1 shows a uniform lamina ABCDE such that ABDE is a rectangle, BC = CD, AB = 4 a and AE = 2 a. The point F is the midpoint of BD and FC = a.

Find, in terms of a, the distance of the centre of mass of the lamina from AE.

The lamina is freely suspended from A and hangs in equilibrium.

Find the angle between AB and the downward vertical.

A uniform rod AB, of mass 3 m and length 4 a, is held in a horizontal position with the end A against a rough vertical wall. One end of a light inextensible string BD is attached to the rod at B and the other end of the string is attached to the wall at the point D vertically above A, where AD = 3 a. A particle of mass 3 m is attached to the rod at C, where AC = x. The rod is in equilibrium in a vertical plane perpendicular to the wall as shown in Figure 3. The

Show that

x = 3 a,

the horizontal component of the force exerted by the wall on the rod has magnitude 5 mg.

The coefficient of friction between the wall and the rod is —. Given that the rod is about to slip,

find the value of —.

The uniform L-shaped lamina ABCDEF, shown in Figure 2, has sides AB and FE parallel, and sides BC and ED parallel. The pairs of parallel sides are 9 cm apart. The points A, F, D and C lie on a straight line. AB = BC = 36 cm, FE = ED = 18 cm. ABC = FED = 90°, and BCD = EDF = EFD = BAC = 45°.

Find the distance of the centre of mass of the lamina from

side AB,

side BC.

The lamina is freely suspended from A and hangs in equilibrium.

Find, to the nearest degree, the size of the angle between AB and the vertical.

A uniform plank AB, of weight 100 N and length 4 m, rests in equilibrium with the end A 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, where AC = 3 m, as shown in Figure 4. The plank is resting in a vertical plane which is perpendicular to the axis of the drum, at an angleto the horizontal, where sin = The coefficient of friction between the plank and the ground is . Modelling the plank as a rod, find the least possible value of .

Figure 2 shows a uniform rod AB of mass m and length 4 a. The end A of the rod is freely hinged to a point on a vertical wall. A particle of mass m is attached to the rod at B. One end of a light inextensible string is attached to the rod at C, where AC = 3 a. The other end of the string is attached to the wall at D, where AD = 2 a and D is vertically above A. The rod rests horizontally in equilibrium in a vertical plane perpendicular to the wall and the tension in the string is T.

Show that T = mg√13.

The particle of mass m at B is removed from the rod and replaced by a particle of mass M which is attached to the rod at B. The string breaks if the tension exceeds 2mg√13. Given that the string does not break,

A uniform rod AB, of mass 20 kg and length 4 m, rests with one end A on rough horizontal ground. The rod is held in limiting equilibrium at an angle α to the horizontal, where tan =αα by a force acting at B, as shown in Figure 2. The line of action of this force lies in the vertical plane which contains the rod. The coefficient of friction between the ground and the rod is 0.5. Find the magnitude of the normal reaction of the ground on the rod at A.

[The centre of mass of a semi-circular lamina of radius r is from the centre]

A template T consists of a uniform plane lamina PQROS, as shown in Figure 3. The lamina is bounded by two semicircles, with diameters SO and OR, and by the sides SP, PQ and QR of the rectangle PQRS. The point O is the mid-point of SR, PQ = 12 cm and QR = 2 x cm.

The template T is freely suspended from the point P and hangs in equilibrium. Given that x = 2 and that θ is the angle that PQ makes with the horizontal,

A uniform rod AB, of length 1.5 m and mass 3 kg, is smoothly hinged to a vertical wall at A. The rod is held in equilibrium in a horizontal position by a light strut CD as shown in Figure 1. The rod and the strut lie in the same vertical plane, which is perpendicular to the wall. The end C of the strut is freely jointed to the wall at a point 0.5 m vertically below A. The end D is freely joined to the rod so that AD is 0.5 m.

Find the thrust in CD.

Find the magnitude and direction of the force exerted on the rod AB at A.

A shop sign ABCDEFG is modelled as a uniform lamina, as illustrated in Figure 2. ABCD is a rectangle with BC = 120 cm and DC = 90 cm. The shape EFG is an isosceles triangle with EG = 60 cm and height 60 cm. The mid-point of AD and the mid-point of EG coincide.

Find the distance of the centre of mass of the sign from the side AD.

The sign is freely suspended from A and hangs at rest.

Find the size of the angle between AB and the vertical.

Figure 2

A plank rests in equilibrium against a fixed horizontal pole. The plank is modelled as a uniform rod AB and the pole as a smooth horizontal peg perpendicular to the vertical plane containing AB. The rod has length 3 a and weight W and rests on the peg at C, where AC = 2 a. The end A of the rod rests on rough horizontal ground and AB makes an angle with the ground, as shown in Figure 2.

Show that the normal reaction on the rod at A is 4(4(4 .32cos α W)32cos cos² ) W Given that the rod is in limiting equilibrium and that cos α =cos =

find the coefficient of friction between the rod and the ground.

Figure 3

Figure 3 shows a rectangular lamina OABC. The coordinates of O, A, B and C are (0, 0), (8, 0), (8, 5) and (0, 5) respectively. Particles of mass km, 5 m and 3 m are attached to the lamina at A, B and C respectively. The x-coordinate of the centre of mass of the three particles without the lamina is 6.4.

Show that k = 7.

The lamina OABC is uniform and has mass 12 m.

Find the coordinates of the centre of mass of the combined system consisting of the three particles and the lamina.

The combined system is freely suspended from O and hangs at rest.

Find the angle between OC and the horizontal.

A set square S is made by removing a circle of centre O and radius 3 cm from a triangular piece of wood. The piece of wood is modelled as a uniform triangular lamina ABC, with ABC = 90°, AB = 12 cm and BC = 21 cm. The point O is 5 cm from AB and 5 cm from BC, as shown in Figure 1.

Find the distance of the centre of mass of S from

AB,

BC.

The set square is freely suspended from C and hangs in equilibrium.

Find, to the nearest degree, the angle between CB and the vertical.

A ladder AB, of mass m and length 4 a, has one end A resting on rough horizontal ground. The other end B rests against a smooth vertical wall. A load of mass 3 m is fixed on the ladder at the point C, where AC = a. The ladder is modelled as a uniform rod in a vertical plane perpendicular to the wall and the load is modelled as a particle. The ladder rests in limiting equilibrium making an angle of 30° with the wall, as shown in Figure 2. Find the coefficient of friction between the ladder and the ground.