Centres of mass

A uniform rod is cut into six pieces. The pieces are used to form the framework ABCDE shown in Figure 1.

• All the pieces of the framework lie in the same plane.

• AB = BC = CD = 2a

• Point E is the midpoint of AC

• Angle ABC = angle BCD = 90°

The distance of the centre of mass of the framework from AB is d

The mass of the framework is M. A particle of mass kM is attached to the framework at B

The distance of the centre of mass of the loaded framework from AB is a

Find the value of k

The shaded region shown in Figure 2 is bounded by the curve with equation y = , the line with equation x = 2, the line with equation x = 4, and the x-axis. This region is rotated through 360° about the x-axis to form a solid.

This solid is used to model a uniform solid pedestal of height 2 m, upper radius 0.5 m and base radius 0.25 m.

Given that the volume of the pedestal is

show that the centre of mass of the pedestal is (4 – 4 ln 2) m from its base.

The pedestal is placed on a rough plane that is inclined at an angle α° to the horizontal. The plane base of the pedestal is in contact with the inclined plane, as shown in Figure 3. The inclined plane is sufficiently rough to prevent the pedestal from sliding.

Given that the pedestal is on the point of toppling,

find the value of α

The uniform rectangular lamina ACDF shown in Figure 5 has AC = FD = 15a and AF = CD = 3 a. The point B on AC is such that AB = 6 a. The point E on FD is such that ED = 6 a.

The rectangular lamina is folded along BE to form the folded lamina shown in Figure 6. The folded lamina has AB perpendicular to BC and the two sections, ABEF and BEDC, of the lamina lie in the same plane.

Show that the distance of the centre of mass of the folded lamina from EF is a.

Explain why the centre of mass of the folded lamina lies on the perpendicular bisector of BE.

The folded lamina is freely suspended from F and hangs in equilibrium.

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

A uniform semicircular lamina has radius r and the midpoint of the diameter of the lamina is at O. The distance of the centre of mass of the lamina from O is d.

Use algebraic integration to show that d =

The uniform plane template T, shown shaded in Figure 7, is formed from a rectangle and a semicircular ring. The rectangle has AD = BC = 2a and AB = DC = 8a. The midpoint of CD is O. The semicircular ring has centre O, outer radius OD = 4a and inner radius OE = 2a. The template is modelled as a uniform lamina.

The template is freely suspended from A. The weight of T is W newtons. A horizontal force is applied to T at B so that T is held in equilibrium with AB vertical. The force acts in the same vertical plane as T and has magnitude λW newtons.

Find the value of λ

A uniform rod of length 28a is cut into seven identical rods each of length 4a. These rods are joined together to form the rigid framework ABCDEA shown in Figure 1.

All seven rods lie in the same plane.

The distance of the centre of mass of the framework from ED is d.

The weight of the framework is W.

The framework is freely pivoted about a horizontal axis through C.

The framework is held in equilibrium in a vertical plane, with AC vertical and A below C, by a horizontal force that is applied to the framework at A.

The force acts in the same vertical plane as the framework and has magnitude F.

Find F in terms of W.

A uniform lamina OAB is in the shape of the region R. Region R lies in the first quadrant and is bounded by the curve with equation 1, the x‑axis, and the y‑axis, as shown shaded in Figure 3.

The point A is the point of intersection of the curve and the x‑axis. The point B is the point of intersection of the curve and the y ‑axis.

One unit on each axis represents 1 m.

The area of R is 6π The centre of mass of R lies at the point with coordinates

Use algebraic integration to show that

Use algebraic integration to find the exact value of

The lamina is freely suspended from A and hangs in equilibrium with OA at angle θ to the downward vertical.

Find the value of θ

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The shaded region, shown in Figure 4, is bounded by the x‑axis, the line with equation x = 6, the line with equation y = 2 and the y ‑axis.

This region is rotated through 360 about the x‑axis to form a solid of revolution. This solid is used to model a non‑uniform cylinder of height 6 cm and radius 2 cm.

The mass per unit volume of the cylinder at the point (x, y, z) is λ(x + 2) kg cm^–3, where 0 x 6 and λ is a constant.

Show that the mass of the cylinder is 120λπkg.

Show that the centre of mass of the cylinder is 3.6 cm from O.

The point O is the centre of one end of the cylinder. The point A is the centre of the other end of the cylinder.

A uniform solid hemisphere of radius 3 cm has density λkg cm^–3. The hemisphere is attached to the cylinder with the centre of its circular face in contact with the point A on the cylinder to form the model shown in Figure 5.

The model is placed with the end containing O on a rough inclined plane which is inclined at angle α to the horizontal. The plane is sufficiently rough to prevent the model from sliding. The model is on the point of toppling.

Find the value of α.

Three particles of masses 3m, 4m and km are positioned at the points with coordinates (2a, 3a), (a, 5a) and (2μa, μa) respectively, where k and μ are constants.

The centre of mass of the three particles is at the point with coordinates (2a, 4a).

Find (i) the value of k

(ii) the value of μ

[ In this question you may quote, without proof, the formula for the distance of the centre of mass of a uniform circular arc from its centre.]

Five pieces of a uniform wire are joined together to form the rigid framework OABCO shown in Figure 1, where

• OA, OB and BC are straight, with OA = OB = BC r

• arc AB is one quarter of a circle with centre O and radius r

• arc OC is one quarter of a circle of radius r

• all five pieces of wire lie in the same plane

Show that the centre of mass of arc AB is a distance from OA.

Given that the distance of the centre of mass of the framework from OA is d,

The framework is freely pivoted at A.

The framework is held in equilibrium, with AO vertical, by a horizontal force of magnitude F which is applied to the framework at C.

Given that the weight of the framework is W

find F in terms of W

A uniform lamina OAB is modelled by the finite region bounded by the x-axis, the y-axis and the curve with equation y 9 x², for x0, as shown shaded in Figure 3. The unit of length on both axes is 1 m.

The area of the lamina is 18 m²

Show that the centre of mass of the lamina is 3.6 m from OB.

[ Solutions relying on calculator technology are not acceptable.]

A light string has one end attached to the lamina at O and the other end attached to the ceiling. A second light string has one end attached to the lamina at A and the other end attached to the ceiling. The lamina hangs in equilibrium with the strings vertical and OA horizontal. The weight of the lamina is W The tension in the string attached to the lamina at A is λW

Find the value of λ

The shaded region shown in Figure 5 is bounded by the line with equation x a and the curve with equation x² y² 4a²

This shaded region is rotated through 180 about the x-axis to form a solid of revolution.

The dome is modelled as being non-uniform with the mass per unit volume of the dome at the point x, y, z equal to kg m³, where a x2a and λ is a constant.

Show that the distance of the centre of mass of the dome from the centre of its

surface of the cone, as shown in Figure 6.

The weight of the cone is kW and the weight of the dome is 2W

The centre of mass of the toy is a distance d metres from the plane face of the dome.

The toy is suspended from a point on the circumference of the plane face of the dome and hangs freely in equilibrium with the plane face of the dome at an angle α to the downward vertical.

find the exact value of k.

Three particles of masses 2m, 3m and km are placed at the points with coordinates (3 a, 2 a), (a, –4 a) and (–3 a, 4 a) respectively. The centre of mass of the three particles lies at the point with coordinates (, ).

Find in terms of a and k

Find in terms of a and k

Given that the distance of the centre of mass of the three particles from the point (0, 0)

find the possible values of k

Nine uniform rods are joined together to form the rigid framework ABCDEFA, with AB = BC = DF = 3 a, BF = CD = DE = 4 a and AF = FE = CF = 5 a, as shown in Figure 1. All nine rods lie in the same plane.

The mass per unit length of each of the rods BF, CF and DF is twice the mass per unit length of each of the other six rods.

Find the distance of the centre of mass of the framework from AC

The mass of the framework is M. A particle of mass kM is attached to the framework at E to form a loaded framework.

When the loaded framework is freely suspended from F, it hangs in equilibrium with CE horizontal.

Find the exact value of k

The uniform plane lamina shown in Figure 3 is formed from two squares, ABCO and ODEF, and a sector ODC of a circle with centre O. Both squares have sides of length 3 a and AO is perpendicular to OF. The radius of the sector is 3 a

[In part (a) you may use, without proof, any of the centre of mass formulae given in the formulae booklet.] (a) Show that the distance of the centre of mass of the sector ODC from OC is

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

The lamina is freely suspended from F and hangs in equilibrium with FC at an angle θ to the downward vertical.

Find the value of θ

The shaded region shown in Figure 4 is bounded by the x‑axis, the line with equation x = 9 and the line with equationy = x. This shaded region is rotated through 360 about the x‑axis to form a solid of revolution. This solid of revolution is used to model a solid right circular cone of height 9 cm and base radius 3 cm.

The cone is non‑uniform and the mass per unit volume of the cone at the point (x, y, z) is λxkg cm^–3, where 0 ≤ x 9 and λ is constant.

Find the distance of the centre of mass of the cone from its vertex.

A toy is made by joining the circular plane face of the cone to the circular plane face of a uniform solid hemisphere of radius 3 cm, so that the centres of the two plane surfaces coincide.

The weight of the cone is W newtons and the weight of the hemisphere is kW newtons.

When the toy is placed on a smooth horizontal plane with any point of the curved surface of the hemisphere in contact with the plane, the toy will remain at rest.

Find the value of k

A letter P from a shop sign is modelled as a uniform plane lamina which consists of a rectangular lamina, OABDE, joined to a semicircular lamina, BCD, along its diameter BD.

OA = ED = a, AB = 2 a, OE = 4 a, and the diameter BD = 2 a, as shown in Figure 1.

Using the model,

find, in terms of π and a, the distance of the centre of mass of the letter P,

from (i) OE

OA

The letter P is freely suspended from O and hangs in equilibrium. The angle between OE and the downward vertical is α.

Using the model,

find the exact value of tan α

A uniform solid hemisphere H has radius 2a. A solid hemisphere of radius a is removed from the hemisphere H to form a bowl. The plane faces of the hemispheres coincide and the centres of the two hemispheres coincide at the point O, as shown in Figure 2.

The centre of mass of the bowl is at the point G.

Figure 3 below shows a cross-section of the bowl which is resting in equilibrium with a point P on its curved surface in contact with a rough plane. The plane is inclined to the horizontal at an angle α and is sufficiently rough to prevent the bowl from slipping. The line OG is horizontal and the points O, G and P lie in a vertical plane which passes through a line of greatest slope of the inclined plane.

Find the size of α, giving your answer in degrees to 3 significant figures.

[In this question, you may assume that the centre of mass of a circular arc, radius r, with angle at centre 2α, is a distance from the centre.]

A thin non-uniform metal plate is in the shape of a sector OAB of a circle with centre O and

radius a. The angle AOB =

The plate is modelled as a non-uniform lamina.

The mass per unit area of the lamina, at any point P of the lamina, is modelled as

Using the model,

find the mass of the plate in terms of λ,

find, in terms of a, the distance of the centre of mass of the plate from O.

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 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.