Class 11th

$x=5sin\left(\pi t+\frac{\pi }{3}\right)m$

Amplitude  $=5 m$

$=5 m$

$\omega =\pi =\frac{2\pi }{T}$

$T=\frac{2\pi }{\pi }=2 s$

Moment of inertia of $rod=I=\frac{m{\mathcal{l}}^2}{12}$

$\Rightarrow 2400=400\frac{{\mathcal{l}}^2}{12}$

$\Rightarrow 72={\mathcal{l}}^2$

$\Rightarrow \mathcal{l}=\sqrt[]{72}=8.48 cm=8.5 cm$

Solid angle $d\Omega =\frac{dA}{r^2}$ has dimensions $\left[M^0 L^0 T^0\right]$

Strain  $=\frac{\Delta l}{l}$ has dimensions $\left[M^0 L^0 T^0\right]$

Angle measured in radians is also dimensionless  $\left[M^0 L^0 T^0\right]$

$\theta =\frac{l}{r}$

Resonance occurs when the frequency of an external periodic force matches the natural frequency of a system. From that, physicists know that resonance causes the amplitude of oscillations to increase significantly. This can be beneficial in devices, such as musical instruments, but dangerous in structures like bridges.

The phase in SHM tells us the position and direction of motion of the particle at a specific instant. It determines the state of oscillation and includes both displacement and time information.

The restoring force in SHM is the force that always acts towards the mean position and is directly proportional to the displacement from it. It follows F=? kx. Here, the negative sign indicates the force is in the opposite direction to the displacement.

Sol. Before collision $\Rightarrow$

It undergoes completely inelastic collision

Using conservation of linear momentum

Initial momentum = Final momentum

$\Rightarrow m{v}_1=m{v}_2+m{v}_2$

$\Rightarrow m{v}_1=2m{v}_2$

$\Rightarrow \frac{v_1}{v_2}=\frac{2}{1}$

$x=2t-1$

$v=\frac{dx}{dt}=2 m s^{-1}$

$P=F·v$

$=2*5=10 W$

In the case of pure rolling,

The topmost point will have velocity $2v$ while point $Q$ i.e. the lowest point will have zero velocity. Hence point $P$ moves faster than point $Q$ .

Air resistance resists the motion of an object. In this case, the net acceleration is lesser than 'g' and it shrinks as the speed increases. This makes the object to speed up more slowly. Ultimately, it reaches a constant terminal velocity which is lower for large-area ones and higher for heavy and streamlined ones.