CAIE Further Paper 3 2022 June — Question 3 4 marks

Exam BoardCAIE
ModuleFurther Paper 3 (Further Paper 3)
Year2022
SessionJune
Marks4
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Mark schemeDownload PDF ↗
TopicVariable acceleration (1D)
TypeDisplacement from velocity by integration
DifficultyStandard +0.3 This is a standard non-constant acceleration problem requiring integration of a(t) to find v(t), then integration again to find x(t). The acceleration function 4000/(5t+4)³ integrates cleanly using substitution (u=5t+4), and initial conditions are clearly stated. This is a routine Further Maths mechanics exercise with straightforward calculus—slightly easier than average A-level difficulty due to its mechanical nature.
Spec1.08f Area between two curves: using integration1.08k Separable differential equations: dy/dx = f(x)g(y)6.06a Variable force: dv/dt or v*dv/dx methods
3 A particle \(P\) is moving in a horizontal straight line. Initially \(P\) is at the point \(O\) on the line and is moving with velocity \(25 \mathrm {~ms} ^ { - 1 }\). At time \(t \mathrm {~s}\) after passing through \(O\), the acceleration of \(P\) is \(\frac { 4000 } { ( 5 t + 4 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) in the direction \(P O\). The displacement of \(P\) from \(O\) at time \(t\) is \(x \mathrm {~m}\). Find an expression for \(x\) in terms of \(t\). \includegraphics[max width=\textwidth, alt={}, center]{c486c59a-2493-4dd3-bf1e-dde57fe744d9-06_894_809_260_628} An object is composed of a hemispherical shell of radius \(2 a\) attached to a closed hollow circular cylinder of height \(h\) and base radius \(a\). The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. \(A B\) is a diameter of the lower end of the cylinder (see diagram).
  1. Find, in terms of \(a\) and \(h\), an expression for the distance of the centre of mass of the object from \(A B\). [4]
    The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 2 } { 3 }\). The object is in equilibrium with \(A B\) in contact with the plane and lying along a line of greatest slope of the plane.
  2. Find the set of possible values of \(h\), in terms of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{c486c59a-2493-4dd3-bf1e-dde57fe744d9-08_629_1358_269_367} A light inextensible string \(A B\) passes through two small holes \(C\) and \(D\) in a smooth horizontal table where \(A C = 3 a\) and \(D B = a\). A particle of mass \(m\) is attached at the end \(A\) and moves in a horizontal circle with angular velocity \(\omega\). A particle of mass \(\frac { 3 } { 4 } m\) is attached to the end \(B\) and moves in a horizontal circle with angular velocity \(k \omega\). \(A C\) makes an angle \(\theta\) with the downward vertical and \(D B\) makes an angle \(\theta\) with the horizontal (see diagram). Find the value of \(k\).
Question 3:
AnswerMarks Guidance
\(\frac{dv}{dt} = \frac{4000}{5t^4} - \frac{400}{5t^3}\)M1 A1 Integrate. Constant of integration needed for A1.
\(t = 0, v = 25\): \(A = 25250\)M1 Find constant.
\(v = \frac{dx}{dt}\): \(x = 80 - \frac{5t^4}{1}\)B1
\(x = 0, t = 0\): \(B = 20\)M1 Integrate and find constant.
\(x = 20 - \frac{80}{5t^4} + \frac{100t}{5t^4}\)A1 
**Question 3:**

$\frac{dv}{dt} = \frac{4000}{5t^4} - \frac{400}{5t^3}$ | M1 A1 | Integrate. Constant of integration needed for A1.

$t = 0, v = 25$: $A = 25250$ | M1 | Find constant.

$v = \frac{dx}{dt}$: $x = 80 - \frac{5t^4}{1}$ | B1 | 

$x = 0, t = 0$: $B = 20$ | M1 | Integrate and find constant.

$x = 20 - \frac{80}{5t^4} + \frac{100t}{5t^4}$ | A1 |
3 A particle $P$ is moving in a horizontal straight line. Initially $P$ is at the point $O$ on the line and is moving with velocity $25 \mathrm {~ms} ^ { - 1 }$. At time $t \mathrm {~s}$ after passing through $O$, the acceleration of $P$ is $\frac { 4000 } { ( 5 t + 4 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }$ in the direction $P O$. The displacement of $P$ from $O$ at time $t$ is $x \mathrm {~m}$.

Find an expression for $x$ in terms of $t$.\\

\includegraphics[max width=\textwidth, alt={}, center]{c486c59a-2493-4dd3-bf1e-dde57fe744d9-06_894_809_260_628}

An object is composed of a hemispherical shell of radius $2 a$ attached to a closed hollow circular cylinder of height $h$ and base radius $a$. The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. $A B$ is a diameter of the lower end of the cylinder (see diagram).
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $a$ and $h$, an expression for the distance of the centre of mass of the object from $A B$. [4]\\

The object is placed on a rough plane which is inclined to the horizontal at an angle $\theta$, where $\tan \theta = \frac { 2 } { 3 }$. The object is in equilibrium with $A B$ in contact with the plane and lying along a line of greatest slope of the plane.
\item Find the set of possible values of $h$, in terms of $a$.\\

\includegraphics[max width=\textwidth, alt={}, center]{c486c59a-2493-4dd3-bf1e-dde57fe744d9-08_629_1358_269_367}

A light inextensible string $A B$ passes through two small holes $C$ and $D$ in a smooth horizontal table where $A C = 3 a$ and $D B = a$. A particle of mass $m$ is attached at the end $A$ and moves in a horizontal circle with angular velocity $\omega$. A particle of mass $\frac { 3 } { 4 } m$ is attached to the end $B$ and moves in a horizontal circle with angular velocity $k \omega$. $A C$ makes an angle $\theta$ with the downward vertical and $D B$ makes an angle $\theta$ with the horizontal (see diagram).

Find the value of $k$.
\end{enumerate}

\hfill \mbox{\textit{CAIE Further Paper 3 2022 Q3 [4]}}
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