Edexcel A-Level Mathematics A2 M3.3 Further Dynamics Questions

Practise further dynamics by deriving force equations, integrating variable-force motion and using SHM tests for speeds, periods and validity.

Exam points

Integrate v dv/dx equations to obtain v² expressions and use boundary conditions.
Derive ẍ=-ω²x for SHM, then use amplitude, speed and period formulae.
Check elastic oscillations stay valid by comparing amplitude with the slack-string condition.

Covered syllabus points

M3.3.1 - Variable-force motion in one dimension M3.3.2 - Simple harmonic motion M3.3.3 - Oscillations attached to elastic strings or springs

[Maximum number: 8]

A particle P of mass $m \mathrm{~kg}$ is initially held at rest at the point O on a smooth inclined plane. The plane is inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha=\frac{2}{5}$

The particle is released from rest and slides down the plane against a force which acts towards O. The force has magnitude $\frac{1}{3} m x^{2} \mathrm{~N}$ , where x metres is the distance of P from O.

(a)

Find the speed of P when x=2

The particle first comes to instantaneous rest at the point A.

[ 6 ]

Using the equation of motion along the plane:
$mv\frac{dv}{dx}=mg\sin\alpha-\frac13mx^2$ $\frac12v^2=xg\sin\alpha-\frac19x^3$

At x=2:
$\frac12v^2=2g\sin\alpha-\frac89$

Since $\sin\alpha=\frac25$ ,
$v=3.7\ \text{or}\ 3.73\text{ m s}^{-1}$

M1 A1 DM1 A1

Alternative energy method:
$mg\sin\alpha\,x=\int\frac13mx^2\,dx+\frac12mv^2$

Marking guidance for this part:
M1 for attempting an equation of motion parallel to the plane.
A1 for correct integration.
DM1 for substituting x=2 into the expression for $v^2$ .
A1 for 3.7 or $3.73\text{ m s}^{-1}$ .

(b)

Find the distance OA.

Figure for Question (b) — Edexcel A-Level Mathematics A2

[ 2 ]

$v=0\Rightarrow x^2=9g\sin\alpha=9g\times\frac25,\qquad x\ne0$ $x=5.939\ldots$ $OA=5.9\text{ or }5.94\text{ m}$
Marking guidance for this part:
M1 for using v=0 in their expression for v and obtaining a value of x.
A1 for 5.9 or $5.94\text{ m}$ .

[Maximum number: 10]

The fixed point A is vertically above the fixed point B, with A B=3 l

A light elastic string has natural length l and modulus of elasticity 4 m g One end of the string is attached to A and the other end is attached to a particle P of mass m

A second light elastic string also has natural length l and modulus of elasticity 4 m g One end of this string is attached to P and the other end is attached to B.

Initially P rests in equilibrium at the point E, where AEB is a vertical straight line.

(a)

Show that $\ddot{x}=-\frac{8 g}{l} x$

[ 4 ]

Top string tension = $4mg\left(\frac{5}{8}+\frac{x}{l}\right)$ , bottom string tension = $4mg\left(\frac{3}{8}-\frac{x}{l}\right)$

mg+4mg\left(\frac{3}{8}-\frac{x}{l}\right)-4mg\left(\frac{5}{8}+\frac{x}{l}\right)=m\ddot{x}

m\ddot{x}=-\frac{8mg}{l}x

A1*

\ddot{x}=-\frac{8g}{l}x

(b)

Find, in terms of g and l, the speed of P when it is $\frac{1}{8} l$ below E.

[ 3 ]

$v=\sqrt{gl}$

(c)

Find the length of time, in each complete oscillation, for which P is more than 1.5 l from A, giving your answer in terms of g and l

[ 3 ]

$t=\sqrt{\frac{l}{2g}}\cos^{-1}\left(-\frac{1}{3}\right)$

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question (a)

Question (b)

Question (c)