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 rod AB has length 8 a and weight W.
The end A of the rod is freely hinged to horizontal ground.
The rod rests in equilibrium against a block which is also fixed to the ground.
The block is modelled as a smooth solid hemisphere with radius 2 a and centre D.
The point of contact between the rod and the block is C, where A C=5 a
The rod is at an angle $\theta$ to the ground, as shown in Figure 1.
Points A, B, C and D all lie in the same vertical plane.

(a)

Show that the magnitude of the normal reaction at C between the rod and the block is $\frac{4}{\sqrt{29}} W$

The resultant force acting on the rod at A has magnitude k W and acts at an angle $\alpha$ to the ground.

[ 3 ]

$\mathrm{M}(A):\ W\times4a\cos\theta=N\times5a$
M1: dimensionally correct equation in a. Allow if a cancelled. Condone sin/cos confusion.
$W\times4a\times\frac{5}{\sqrt{29}}=N\times5a$
A1: correct unsimplified equation in a. Allow with $\cos\theta$ .
$N=\frac{4}{\sqrt{29}}W\quad *$
A1*: obtain given answer from correct working.
[3]

(b)

(c)

the exact value of $\tan \alpha$

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question (a)

Question (b)

Question (c)