Vertical SHM with two strings - Simple Harmonic Motion

Edexcel M3 2021 January Q5

13 marks Challenging +1.2

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-16_720_232_251_858} \captionsetup{labelformat=empty} \caption{Figure 6}

\end{figure} The fixed points, $A$ and $B$, are a distance $10 a$ apart, with $B$ vertically above $A$. One end of a light elastic string, of natural length $2 a$ and modulus of elasticity $2 m g$, is attached to a particle $P$ of mass $m$ and the other end is attached to $A$. One end of another light elastic string, of natural length $4 a$ and modulus of elasticity $6 m g$, is attached to $P$ and the other end is attached to $B$. The particle $P$ rests in equilibrium at the point $C$, as shown in Figure 6.

Show that each string has an extension of $2 a$.
(5) The particle $P$ is now pulled down vertically, so that it is a distance $a$ below $C$ and then released from rest.
Show that in the subsequent motion, $P$ performs simple harmonic motion.
Find, in terms of $a$ and $g$, the speed of $P$ when it is a distance $\frac { 7 } { 2 } a$ above $A$.

Edexcel M3 2024 January Q6

14 marks Challenging +1.2

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 $A E B$ is a vertical straight line.

Show that $A E = \frac { 13 } { 8 } l$ The particle $P$ is now held at the point that is a distance $2 l$ vertically below $A$ and released from rest. At time $t$, the vertical displacement of $P$ from $E$ is $x$, where $x$ is measured vertically downwards.
Show that $\ddot { x } = - \frac { 8 g } { l } x$
Find, in terms of $g$ and $l$, the speed of $P$ when it is $\frac { 1 } { 8 } l$ below $E$.
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 $l$

OCR MEI M3 2006 June Q3

18 marks Challenging +1.2

3 A fixed point A is 12 m vertically above a fixed point B. A light elastic string, with natural length 3 m and modulus of elasticity 1323 N , has one end attached to A and the other end attached to a particle P of mass 15 kg . Another light elastic string, with natural length 4.5 m and modulus of elasticity 1323 N , has one end attached to B and the other end attached to P .

Verify that, in the equilibrium position, $\mathrm { AP } = 5 \mathrm {~m}$. The particle P now moves vertically, with both strings AP and BP remaining taut throughout the motion. The displacement of P above the equilibrium position is denoted by $x \mathrm {~m}$ (see Fig. 3). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-4_405_360_751_849} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
Show that the tension in the string AP is $441 ( 2 - x ) \mathrm { N }$ and find the tension in the string BP .
Show that the motion of P is simple harmonic, and state the period. The minimum length of AP during the motion is 3.5 m .
Find the maximum length of AP .
Find the speed of P when $\mathrm { AP } = 4.1 \mathrm {~m}$.
Find the time taken for AP to increase from 3.5 m to 4.5 m .

Edexcel M4 2010 June Q6

17 marks Challenging +1.2

Two points $A$ and $B$ lie on a smooth horizontal table with $A B = 4 a$. One end of a light elastic spring, of natural length $a$ and modulus of elasticity $2 m g$, is attached to $A$. The other end of the spring is attached to a particle $P$ of mass $m$. Another light elastic spring, of natural length $a$ and modulus of elasticity $m g$, has one end attached to $B$ and the other end attached to $P$. The particle $P$ is on the table at rest and in equilibrium.
1. Show that $A P = \frac { 5 a } { 3 }$.
The particle $P$ is now moved along the table from its equilibrium position through a distance $0.5 a$ towards $B$ and released from rest at time $t = 0$. At time $t , P$ is moving with speed $v$ and has displacement $x$ from its equilibrium position. There is a resistance to motion of magnitude $4 m \omega v$ where $\omega = \sqrt { } \left( \frac { g } { a } \right)$.
Show that $\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + 3 \omega ^ { 2 } x = 0$.
Find the velocity, $\frac { \mathrm { d } x } { \mathrm {~d} t }$, of $P$ in terms of $a , \omega$ and $t$.

Edexcel M4 2012 June Q6

11 marks Challenging +1.2

6. Two points $A$ and $B$ are in a vertical line, with $A$ above $B$ and $A B = 4 a$. One end of a light elastic spring, of natural length $a$ and modulus of elasticity $3 m g$, is attached to $A$. The other end of the spring is attached to a particle $P$ of mass $m$. Another light elastic spring, of natural length $a$ and modulus of elasticity $m g$, has one end attached to $B$ and the other end attached to $P$. The particle $P$ hangs at rest in equilibrium.

Show that $A P = \frac { 7 a } { 4 }$ The particle $P$ is now pulled down vertically from its equilibrium position towards $B$ and at time $t = 0$ it is released from rest. At time $t$, the particle $P$ is moving with speed $v$ and has displacement $x$ from its equilibrium position. The particle $P$ is subject to air resistance of magnitude $m k v$, where $k$ is a positive constant.
Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + k \frac { \mathrm {~d} x } { \mathrm {~d} t } + \frac { 4 g } { a } x = 0$$
Find the range of values of $k$ which would result in the motion of $P$ being a damped oscillation.

OCR M4 2008 June Q7

18 marks Challenging +1.8

7
\includegraphics[max width=\textwidth, alt={}, center]{a9e010ce-c3a8-4f95-a154-fd16ef3e5e5b-4_622_767_269_689} Particles $P$ and $Q$, with masses $3 m$ and $2 m$ respectively, are connected by a light inextensible string passing over a smooth light pulley. The particle $P$ is connected to the floor by a light spring $S _ { 1 }$ with natural length $a$ and modulus of elasticity mg . The particle $Q$ is connected to the floor by a light spring $S _ { 2 }$ with natural length $a$ and modulus of elasticity $2 m g$. The sections of the string not in contact with the pulley, and the two springs, are vertical. Air resistance may be neglected. The particles $P$ and $Q$ move vertically and the string remains taut; when the length of $S _ { 1 }$ is $x$, the length of $S _ { 2 }$ is ( $3 a - x$ ) (see diagram).

Find the total potential energy of the system (taking the floor as the reference level for gravitational potential energy). Hence show that $x = \frac { 4 } { 3 } a$ is a position of stable equilibrium.
By differentiating the energy equation, and substituting $x = \frac { 4 } { 3 } a + y$, show that the motion is simple harmonic, and find the period.