Edexcel FM2 AS (Further Mechanics 2 AS) 2020 June

Question 1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0567d068-e23c-446e-9e11-f0c292972093-02_490_824_253_588} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a uniform rectangular lamina $A B C D$ with $A B = 2 a$ and $A D = a$ The mass of the lamina is $6 m$. A particle of mass $2 m$ is attached to the lamina at $A$, a particle of mass $m$ is attached to the lamina at $B$ and a particle of mass $3 m$ is attached to the lamina at $D$, to form a loaded lamina $L$ of total mass $12 m$.

Write down the distance of the centre of mass of $L$ from $A B$. You must give a reason for your answer.
Show that the distance of the centre of mass of $L$ from $A D$ is $\frac { 2 a } { 3 }$ A particle of mass $k m$ is now also attached to $L$ at $D$ to form a new loaded lamina $N$.
Show that the distance of the centre of mass of $N$ from $A B$ is $\frac { ( k + 6 ) a } { ( k + 12 ) }$ When $N$ is freely suspended from $A$ and is hanging in equilibrium, the side $A B$ makes an angle $\alpha$ with the vertical, where $\tan \alpha = \frac { 3 } { 2 }$
Find the value of $k$.

Question 2

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2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0567d068-e23c-446e-9e11-f0c292972093-06_531_837_258_632} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} One end of a string of length $3 a$ is attached to a point $A$ and the other end is attached to a point $B$ on a smooth horizontal table. The point $B$ is vertically below $A$ with $A B = a \sqrt { 3 }$ A small smooth bead, $P$, of mass $m$ is threaded on to the string. The bead $P$ moves on the table in a horizontal circle, with centre $B$, with constant speed $U$. Both portions, $A P$ and $B P$, of the string are taut, as shown in Figure 2. The string is modelled as being light and inextensible and the bead is modelled as a particle.

Show that $A P = 2 a$
Find, in terms of $m , U$ and $a$, the tension in the string.
Show that $U ^ { 2 } < a g \sqrt { 3 }$
Describe what would happen if $U ^ { 2 } > a g \sqrt { 3 }$
State briefly how the tension in the string would be affected if the string were not modelled as being light.

Question 3

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At time $t = 0$, a toy electric car is at rest at a fixed point $O$. The car then moves in a horizontal straight line so that at time $t$ seconds $( t > 0 )$ after leaving $O$, the velocity of the car is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the acceleration of the car is modelled as $( p + q v ) \mathrm { ms } ^ { - 2 }$, where $p$ and $q$ are constants.

When $t = 0$, the acceleration of the car is $3 \mathrm {~ms} ^ { - 2 }$
When $t = T$, the acceleration of the car is $\frac { 1 } { 2 } \mathrm {~ms} ^ { - 2 }$ and $v = 4$

Show that $$8 \frac { \mathrm {~d} v } { \mathrm {~d} t } = ( 24 - 5 v )$$
Find the exact value of $T$, simplifying your answer.