Edexcel IAL Mathematics M3.3 further dynamics

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: 5]

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

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

$0.5 v \frac{\mathrm{~d} v}{\mathrm{~d} x}=-\frac{2}{x^{3}}$

M1

$\int v \mathrm{~d} v=\int-\frac{4}{x^{3}} \mathrm{~d} x$ $\frac{v^{2}}{2}=\frac{2}{x^{2}} \quad(+k)$

DM1A1

$v^{2}=\frac{4}{x^{2}}+2 k$ $v=3, x=1 \Rightarrow 2 k=5$

DM1

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

A1 *

(5)

Alt for first 2 marks using Energy

Use of $W=\int F d x$

M1

Attempt integration and equate to change in KE.

M1

[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 ]

(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 ]

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Covered syllabus points

M3.3 - Further dynamics question 1

M3.3 - Further dynamics question 2

Question (a)

Question (b)

Question (c)