Prove motion is SHM from equation

A question is this type if and only if it gives an equation for displacement (e.g., x = A cos(ωt)) and asks to prove the particle is moving with SHM by showing acceleration satisfies the SHM condition.

5 questions · Moderate -0.1

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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 $B$.

Edexcel M3 2011 January Q4

11 marks Moderate -0.3

A particle $P$ moves along the $x$-axis. At time $t$ seconds its displacement, $x$ metres, from the origin $O$ is given by $x = 5 \sin \left( \frac { 1 } { 3 } \pi t \right)$.
1. Prove that $P$ is moving with simple harmonic motion.
2. Find the period and the amplitude of the motion.
3. Find the maximum speed of $P$.
The points $A$ and $B$ on the positive $x$-axis are such that $O A = 2 \mathrm {~m}$ and $O B = 3 \mathrm {~m}$.
Find the time taken by $P$ to travel 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 $t = 14$.