Standard +0.3 This is a standard dimensional analysis question with routine mechanics calculations. Part (b)(iii) requires setting up three simultaneous equations from dimensional consistency, which is a textbook exercise. The mechanics parts involve straightforward application of formulas (conical pendulum, elastic strings, energy conservation) with no novel problem-solving required. Slightly easier than average due to the guided structure and standard techniques.
A particle P of mass 1.5 kg is connected to a fixed point by a light inextensible string of length 3.2 m . The particle P is moving as a conical pendulum in a horizontal circle at a constant angular speed of \(2.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
Find the tension in the string.
Find the angle that the string makes with the vertical.
A particle Q of mass \(m\) moves on a smooth horizontal surface, and is connected to a fixed point on the surface by a light elastic string of natural length \(d\) and stiffness \(k\). With the string at its natural length, Q is set in motion with initial speed \(u\) perpendicular to the string. In the subsequent motion, the maximum length of the string is \(2 d\), and the string first returns to its natural length after time \(t\).
You are given that \(u = \sqrt { \frac { 4 k d ^ { 2 } } { 3 m } }\) and \(t = A k ^ { \alpha } d ^ { \beta } m ^ { \gamma }\), where \(A\) is a dimensionless constant.
Show that the dimensions of \(k\) are \(\mathrm { MT } ^ { - 2 }\).
Show that the equation \(u = \sqrt { \frac { 4 k d ^ { 2 } } { 3 m } }\) is dimensionally consistent.
Find \(\alpha , \beta\) and \(\gamma\).
You are now given that Q has mass 5 kg , and the string has natural length 0.7 m and stiffness \(60 \mathrm { Nm } ^ { - 1 }\).
Find the initial speed \(u\), and use conservation of energy to find the speed of Q at the instant when the length of the string is double its natural length.
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\begin{enumerate}[label=(\alph*)]
\item A particle P of mass 1.5 kg is connected to a fixed point by a light inextensible string of length 3.2 m . The particle P is moving as a conical pendulum in a horizontal circle at a constant angular speed of $2.5 \mathrm { rad } \mathrm { s } ^ { - 1 }$.
\begin{enumerate}[label=(\roman*)]
\item Find the tension in the string.
\item Find the angle that the string makes with the vertical.
\end{enumerate}\item A particle Q of mass $m$ moves on a smooth horizontal surface, and is connected to a fixed point on the surface by a light elastic string of natural length $d$ and stiffness $k$. With the string at its natural length, Q is set in motion with initial speed $u$ perpendicular to the string. In the subsequent motion, the maximum length of the string is $2 d$, and the string first returns to its natural length after time $t$.
You are given that $u = \sqrt { \frac { 4 k d ^ { 2 } } { 3 m } }$ and $t = A k ^ { \alpha } d ^ { \beta } m ^ { \gamma }$, where $A$ is a dimensionless constant.
\begin{enumerate}[label=(\roman*)]
\item Show that the dimensions of $k$ are $\mathrm { MT } ^ { - 2 }$.
\item Show that the equation $u = \sqrt { \frac { 4 k d ^ { 2 } } { 3 m } }$ is dimensionally consistent.
\item Find $\alpha , \beta$ and $\gamma$.
You are now given that Q has mass 5 kg , and the string has natural length 0.7 m and stiffness $60 \mathrm { Nm } ^ { - 1 }$.
\item Find the initial speed $u$, and use conservation of energy to find the speed of Q at the instant when the length of the string is double its natural length.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI M3 2013 Q1 [18]}}