Edexcel A-Level Mathematics A2 Unit M3 Mechanics 3 Questions

Practise Mechanics 3 topics involving variable acceleration, elastic strings, circular motion, energy methods and rigid body statics.

Covered syllabus points

M3.1 - Further kinematics M3.2 - Elastic strings and springs M3.3 - Further dynamics M3.4 - Motion in a circle M3.5 - Statics of rigid bodies

[Maximum number: 12]

A particle P of mass 0.5 kg moves along the positive x-axis in the positive x direction.

At time t seconds, $t \geqslant 1, P$ is x metres from the origin O and is moving with speed $v \mathrm{~m} \mathrm{~s}^{-1}$ . The resultant force acting on P has magnitude $\frac{2}{x^{3}} \mathrm{~N}$ and is directed towards O.

When t=1, x=1 and v=3

Show that

(a)

$v^{2}=\frac{4}{x^{2}}+5$

[ 5 ]

$0.5 v \frac{\mathrm{~d} v}{\mathrm{~d} x}=-\frac{2}{x^{3}}$ $\int v \mathrm{~d} v=\int-\frac{4}{x^{3}} \mathrm{~d} x$ $\frac{v^{2}}{2}=\frac{2}{x^{2}} \quad(+k)$ $v^{2}=\frac{4}{x^{2}}+2 k$ $v=3, x=1 \Rightarrow 2 k=5$ $v^{2}=\frac{4}{x^{2}}+5 *$

A1 *

(5)

Alt for first 2 marks using Energy

Use of $W=\int F d x$
Attempt integration and equate to change in KE.

(b)

$t=\frac{a+\sqrt{b x^{2}+c}}{d}$ , where a, b, c and d are integers to be found.

[ 7 ]

$\frac{\mathrm{d} x}{\mathrm{~d} t}=\sqrt{\frac{4}{x^{2}}+5}$ $\int \frac{x}{\sqrt{5 x^{2}+4}} \mathrm{~d} x=\int \mathrm{d} t$ $\frac{1}{5} \sqrt{5 x^{2}+4}=t \quad\left(+k_{1}\right)$ $x=1, t=1 \Rightarrow k_{1}=-\frac{2}{5}$ $t=\frac{2+\sqrt{5 x^{2}+4}}{5}$
(7)
(a)

M1 Dimensionally correct equation of motion. Acceleration must be in form $v \frac{\mathrm{~d} v}{\mathrm{~d} x}$ . Condone missing minus sign.

DM1 Separate variables and attempt integration. Condone missing minus sign.
A1 Correct integrals. Condone missing constant.
DM1 Use v=3, x=1 to find constant of integration. Dependent on both previous M marks.
A1* Reach given result with no errors.

Alt - Definite Integration/Energy Work.
M1 - Equate change in KE to Integral for WD. Integration not needed for this mark and condone inconsistent signs.

\frac{1}{2} \times 2 v^{2}-\frac{1}{2} \times 2 \times 3^{2}=\int( \pm) \frac{2}{x^{3}} d x

DM1 Attempt integration. Condone inconsistent signs.
A1 Correct integration, with consistent signs for KE and WD.
DM1 Substitute in limits.
A1 Reach given result with no errors.
(b)

M1 Use $v=\frac{\mathrm{d} x}{\mathrm{~d} t}$ to form differential equation in x and t.
M1 Separate variables and to produce functions ready to be integrated.
A1 Correct integrands, written in a form that can be integrated.
DM1 Valid attempt to integrate their expression. If they have an incorrect expression, the integration must not be significantly simplified. Dependent on first 2 M marks.

A1 Correct integration. Condone missing constant.
DM1 Use x=1, t=1 to find constant of integration. Dependent on all 3 M marks.
A1 Correct result. (cso)

Question
Number

Scheme

Marks

[Maximum number: 4]

Figure 2

A smooth bead of weight 12 N is threaded onto a light elastic string of natural length 3 m. The points A and B are on a horizontal ceiling, with $A B=3 \mathrm{~m}$ . One end of the string is attached to A and the other end of the string is attached to B.

The bead hangs freely in equilibrium, 2 m below the ceiling, as shown in Figure 2.

(a)

Show that the modulus of elasticity of the string is 11.25 N .

The bead is now pulled down to a point vertically below its equilibrium position and released from rest.

[ 2 ]

Ext $=2(2.5)-3=2 \mathrm{~m}$ $7.5=\frac{\lambda \times 2}{3} \Rightarrow \lambda=11.25(\mathrm{~N}) *$

M1A1*

(2)

(b)

Find the elastic energy stored in the string at the instant when the bead is moving at its maximum speed.

[ 2 ]

$\mathrm{EPE}=\frac{11.25 \times 2^{2}}{2 \times 3}=7.5(\mathrm{~J})$
(2)

[8]
(a)

B1 $\cos \theta=\frac{4}{5}$ seen or implied.
M1 Resolving vertically, with 2 equal tensions (implied) and weight.
A1 Correct equation.
A1 $T=\frac{15}{2}=7.5 \mathrm{~N}$ accept any equivalent fraction, since g not used.
(b)

M1 Use of Hooke's Law with their tension to form equation in $\lambda$ . Must be using natural length of 3 (or 1.5 for half string), but condone incorrect extension for M mark.

A1* 11.25 or any equivalent fraction.
(c)

M1 Attempt at EPE in equilibrium position. Must have same extension as (b). Condone missing half in EPE formula. If using half strings, then they must include the EPE of both strings. Must be using natural length of 3 (or 1.5 for half string). Allow if an embedded term in an energy equation.

A1 7.5 J. Accept any equivalent. Must be a clear answer (not embedded).

Question
Number

Scheme

Marks

[Maximum number: 5]

Use algebraic integration to show that the centre of mass of a uniform semicircular disc of radius r and centre O is at a distance $\frac{4 r}{3 \pi}$ from the diameter through O [You may assume, without proof, that the area of a circle of radius r is $\pi r^{2}$ ]

A uniform lamina L is in the shape of a semicircle with centre B and diameter A C=8 a. The semicircle with diameter AB is removed from L and attached to the straight edge BC to form the template T, shown shaded in Figure 4.

Figure 4

The distance of the centre of mass of T from AC is d.

\bar{x}=\frac{\pi\int_0^r x(r^2-x^2)\,dx}{\frac{2}{3}\pi r^3}

=\frac{\pi\left[\frac{r^2x^2}{2}-\frac{x^4}{4}\right]_0^r}{\frac{2}{3}\pi r^3}

=\frac{\pi\cdot\frac{r^4}{4}}{\frac{2}{3}\pi r^3}

A1*

\bar{x}=\frac{4r}{3\pi}

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question (a)

Question (b)

Question 3