Edexcel A-Level Mathematics A2 M2.5 Statics of Rigid Bodies Questions

Practise rigid body statics with rods, beams and girders by taking moments, resolving forces and finding reactions or limiting values.

Exam points

take moments about a chosen point to form equations for rods, beams or girders
resolve coplanar forces to find hinge reactions or resultant forces at supports
use limiting equilibrium conditions to calculate masses, loads or largest possible constants

Covered syllabus points

M2.5.1 - Moment of a force M2.5.2 - Equilibrium of rigid bodies

[Maximum number: 9]

Figure 4

A uniform $\operatorname{rod} A B$ has length 8 a and weight W.
The end A of the rod is freely hinged to a fixed point on a vertical wall.
A particle of weight $\frac{1}{4} W$ is attached to the rod at B.
A light inelastic string of length 5 a has one end attached to the rod at the point C, where A C=5 a.

The other end of the string is attached to the wall at the point D, where D is above A and A D=5 a, as shown in Figure 4.

The rod rests in equilibrium.
The tension in the string is T.

(a)

Show that $T=\frac{6}{5} \mathrm{~W}$

[ 3 ]

Official answer figure

Moments about A
$\begin{gathered} 4a\cos30^\circ\,W+8a\cos30^\circ\left(\frac W4\right) =5a\cos30^\circ\,T \end{gathered}$ $6W=5T\Rightarrow T=\frac65W$

A1*

(b)

Find, in terms of W, the magnitude of the force exerted on the rod by the hinge at A.

[ 6 ]

Use two independent equilibrium equations, for example

$\begin{aligned} T\cos30^\circ(3a)+V\cos30^\circ(8a) &=W\cos30^\circ(4a)+H\sin60^\circ(8a)\\ W\cos30^\circ(a)+H\sin60^\circ(5a) &=\frac14W\cos30^\circ(3a)+V\cos30^\circ(5a) \end{aligned}$
Solve to find the hinge force.

A1ft

[Maximum number: 3]

Figure 1

A uniform circular disc C has centre X and radius R.
A disc with centre Y and radius r, where 0<r<R and X Y=R-r, is removed from C to form the template shown shaded in Figure 1.

The centre of mass of the template is a distance k r from X.

The mass of the template is M.

Find, in terms of M, the mass of the lightest particle that could be attached to the template so that it would hang in equilibrium from P with XY horizontal.

Moments about an axis through P
$M_1gR=Mg\times\frac49r$

or

$M_1gR=Mg\times\frac{16}{45}R$ $M_1=\frac{16}{45}M$

[Maximum number: 9]

Figure 2

A uniform $\operatorname{rod} A B$ has length 4 m and weight 50 N.
The rod has its end A on rough horizontal ground. The rod is held in equilibrium at an angle $\alpha$ to the ground by a light inextensible cable attached to the rod at B, as shown in Figure 2. The cable and the rod lie in the same vertical plane and the cable is perpendicular to the rod. The tension in the cable is T newtons.

Given that $\sin \alpha=\frac{3}{5}$

(a)

show that T=20

Given also that the rod is in limiting equilibrium,

[ 3 ]

Figure for Answer to Question (a) — Edexcel A-Level Mathematics A2

Moments about A.
M1: dimensionally correct equation, force x distance = force x distance. Condone sin/cos confusion. Mark 50g as an accuracy error.
$\begin{array}{r} 4T=2\cos\alpha\times50\\ =2\times\frac45\times50 \end{array}$
A1: correct unsimplified equation. Need to see $\cos\alpha$ or $\frac45$ . Might see LHS $=T\cos\alpha\times4\cos\alpha+T\sin\alpha\times4\sin\alpha$ .
$T=20\quad *$
A1*: obtain given answer from correct working. Must see $\frac45$ used correctly.

(b)

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

[ 6 ]

Resolve horizontally.
M1: condone sin/cos confusion.
$H=T\sin\alpha$
A1: correct equation.
Resolve vertically.
M1: need all 3 terms. Condone sign error and sin/cos confusion.
$T\cos\alpha+V=50$
A1: correct equation.
Either or both of the above equations could be replaced by a moments equation, for example
$M(B):\ 4\cos\alpha\times V=4\sin\alpha\times H+2\cos\alpha\times50$
or by resolving perpendicular and parallel to the rod:
$T+V\cos\alpha=50\cos\alpha+H\sin\alpha$ $50\sin\alpha=H\cos\alpha+V\sin\alpha$
Use $F=\mu R$ to form an equation in $\mu$ .
M1: $H=\mu V$ used, not just stated;
they must get as far as substituting their values.
$\mu=\frac{6}{17}$
A1: $\mu=0.35$ or better. Accept $\frac{12}{34}$ .

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question 3

Question (a)

Question (b)