A particle is in linear simple harmonic motion between two points. $A$ and $B, 10 \mathrm{~cm}$ apart (figure.) take the direction from $A$ to $B$ as the positive direction and choose the correct statements.
AO = OB = 5 cm
BC = 8 cm
Displacement versus time curve for a particle executing SHM is shown in figure. Identify the points marked at which (i) velocity of the oscillator is zero, (ii) speed of the oscillator is maximum.
Displacement versus time curve for a particle executing SHM is shown in figure. Identify the points marked at which (i) velocity of the oscillator is zero, (ii) speed of the oscillator is maximum.
In SHM y-t graph, zero displacement values correspond to mean position; where velocity of the oscillator is maximum.
Whereas the crest and troughs represent extreme positions, where displacement is maximum and velocity of the oscillator is minimum and is zero. Hence,
(a) $A, C, E, G$ are either crest or trough having zero velocity.
(b) speed is maximum at mean positions represented by $B, D, F, H$ points.
Two identical springs of spring constant $k$ are attached to a block of mass $m$ and to fixed supports as shown in figure. When the mass is displaced from equilibrium position by a distance $x$ towards right, find the restoring force.
Consider the diagram in which the block is displaced right through x.
The right spring gets compressed by $x$ developing a restoring force $k x$ towards left on the block. The left spring is stretched by an amount $x$ developing a restoring force $k x$ towards left on the block as given in the free body diagram of the block.
$$\begin{aligned} \text { Hence, total force (restoring) } & =(k x+k x) \quad[\because \text { Both forces are in same direction }] \\ & =2 k x \text { towards left } \end{aligned}$$
What are the two basic characteristics of a simple harmonic motion?
The two basic characteristics of a simple harmonic motion
(i) Acceleration is directly proportional to displacement.
(ii) The direction of acceleration is always towards the mean position, that is opposite to displacement.
When will the motion of a simple pendulum be simple harmonic?
Consider the diagram of a simple pendulum.
The bob is displaced through an angle $\theta$ shown.
The restoring torque about the fixed point $O$ is
$$\tau=-m g \sin \theta$$
If $\theta$ is small angle in radians, then $\sin \theta \approx \theta$
$$\Rightarrow \quad \tau \approx-m g \theta \Rightarrow \tau \propto(-\theta)$$
Hence, motion of a simple pendulum is SHM for small angle of oscillations.