Prove motion is SHM from equation - Simple Harmonic Motion

Edexcel M3 2022 January Q5

12 marks Moderate -0.3

A particle $P$ is moving along the $x$-axis. At time $t$ seconds the displacement of $P$ from the origin $O$ is $x$ metres, where $x = 4 \cos \left( \frac { 1 } { 5 } \pi t \right)$
1. Prove that $P$ is moving with simple harmonic motion.
2. Find the period of the motion.
3. State the amplitude of the motion.
4. Find, in terms of $\pi$, the maximum speed of $P$
The points $A$ and $B$ lie on the $x$-axis, on opposite sides of $O$, with $O A = 1.5 \mathrm {~m}$ and $O B = 2.5 \mathrm {~m}$.
Find the time taken by $P$ to move directly from $A$ to $B$.

OCR MEI M3 2008 January Q3

17 marks Standard +0.3

3 A particle is oscillating in a vertical line. At time $t$ seconds, its displacement above the centre of the oscillations is $x$ metres, where $x = A \sin \omega t + B \cos \omega t$ (and $A , B$ and $\omega$ are constants).

Show that $\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \omega ^ { 2 } x$. When $t = 0$, the particle is 2 m above the centre of the oscillations, the velocity is $1.44 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ downwards, and the acceleration is $0.18 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ downwards.
Find $A , B$ and $\omega$.
Show that the period of oscillation is 20.9 s (correct to 3 significant figures), and find the amplitude.
Find the total distance travelled by the particle between $t = 12$ and $t = 24$.

OCR MEI M3 2010 June Q4

18 marks Standard +0.3

4 A particle P is performing simple harmonic motion in a vertical line. At time $t \mathrm {~s}$, its displacement $x \mathrm {~m}$ above a fixed point O is given by $$x = A \sin \omega t + B \cos \omega t$$ where $A , B$ and $\omega$ are constants.

Show that the acceleration of P , in $\mathrm { ms } ^ { - 2 }$, is $- \omega ^ { 2 } x$. When $t = 0 , \mathrm { P }$ is 16 m below O , moving with velocity $7.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ upwards, and has acceleration $1 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ upwards.
Find the values of $A , B$ and $\omega$.
Find the maximum displacement, the maximum speed, and the maximum acceleration of P .
Find the speed and the direction of motion of P when $t = 15$.
Find the distance travelled by P between $t = 0$ and $t = 15$.

OCR MEI M3 2012 June Q3

18 marks Moderate -0.3

3 A particle Q is performing simple harmonic motion in a vertical line. Its height, $x$ metres, above a fixed level at time $t$ seconds is given by $$x = c + A \cos ( \omega t - \phi )$$ where $c , A , \omega$ and $\phi$ are constants.

Show that $\ddot { x } = - \omega ^ { 2 } ( x - c )$. Fig. 3 shows the displacement-time graph of Q for $0 \leqslant t \leqslant 14$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-4_547_1079_703_495} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
Find exact values for $c , A , \omega$ and $\phi$.
Find the maximum speed of Q .
Find the height and the velocity of Q when $t = 0$.
Find the distance travelled by Q between $t = 0$ and $t = 14$.

WJEC Further Unit 6 2022 June Q2

15 marks Standard +0.8

2. A particle $P$ moves along the $x$-axis such that its position $x$ metres, after $t$ seconds, is given by $$x = \sin ( \pi t ) + \sqrt { 3 } \cos ( \pi t )$$

1. Show that the motion of the particle $P$ is Simple Harmonic. State the value of $x$ at the centre of motion.
2. Show that the period of the motion of $P$ is 2 s and determine the amplitude. Suppose that another particle $Q$ is introduced so that it also moves along the $x$-axis with Simple Harmonic Motion with centre of motion, $O$, and period equal to that of particle $P$. When $t = 0$, the particle $Q$ is at $O$ and when it is $2 \sqrt { 3 } \mathrm {~m}$ from $O$ its speed is $2 \pi \mathrm {~ms} ^ { - 1 }$.
Find the amplitude of particle $Q$.
Determine the time when particles $P$ and $Q$ first meet.

OCR MEI Further Pure Core Specimen Q16

18 marks Challenging +1.2

A small object is attached to a spring and performs oscillations in a vertical line. The displacement of the object at time $t$ seconds is denoted by $x$ cm. Preliminary observations suggest that the object performs simple harmonic motion (SHM) with a period of 2 seconds about the point at which $x = 0$.

1. Write down a differential equation to model this motion. [3]
2. Give the general solution of the differential equation in part (i) (A). [1]

Subsequent observations indicate that the object's motion would be better modelled by the differential equation $$\frac{d^2x}{dt^2} + 2k \frac{dx}{dt} + (k^2 + 9)x = 0 \qquad (*)$$ where $k$ is a positive constant.

1. Obtain the general solution of (*). [3]
2. State two ways in which the motion given by this model differs from that in part (i). [2]

The amplitude of the object's motion is observed to reduce with a scale factor of 0.98 from one oscillation to the next.

Find the value of $k$. [3]

At the start of the object's motion, $x = 0$ and the velocity is 12 cm s$^{-1}$ in the positive $x$ direction.

Find an equation for $x$ as a function of $t$. [4]
Without doing any further calculations, explain why, according to this model, the greatest distance of the object from its starting point in the subsequent motion will be slightly less than 4 cm. [2]