Course Code: BPHCT-131

Assignment Code: BPHCT-131//TMA/2023

Max. Marks: 100

Note: Attempt all questions. The marks for each question are indicated against it.

\section*{PART A}

1. a) Determine two unit vectors perpendicular to both \(\vec{A}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{B}}=-2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}\).

b) For any two vectors \(\overrightarrow{\mathbf{u}}\) and \(\overrightarrow{\mathbf{v}}\) show that:

\[
(\overrightarrow{\mathbf{u}} . \overrightarrow{\mathbf{v}})^{2}-[(\overrightarrow{\mathbf{u}} \times \overrightarrow{\mathbf{v}}) \times \overrightarrow{\mathbf{v}}] . \overrightarrow{\mathbf{u}}=u^{2} v^{2}
\]

2. Solve the following ordinary differential equations:
a) \(\frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}+2 y=0, \quad y(0)=2, \quad y^{\prime}(0)=1\)
b) \(\frac{1}{x} \sin y d x+(\ln x \cos y+y) d y=0\)

3. a) A car has a weight of \(12,000 \mathrm{~N}\). The coefficient of kinetic friction between its wheels and the wet highway is 0.5 . The car is travelling at \(20 \mathrm{~ms}^{-1}\) when brakes are applied. How far does the car travel before it comes to a complete stop? Take \(g=10 \mathrm{~ms}^{-2}\).

b) A crate of mass \(20.0 \mathrm{~kg}\) is pulled by a force of \(180 \mathrm{~N}\), up an inclined plane which makes an angle of \(30^{\circ}\) with the horizon. The coefficient of kinetic friction between the plane and the crate is \(\mu_{\mathrm{k}}=0.225\). If the crates starts from rest, calculate its speed after it has been pulled \(15.0 \mathrm{~m}\). Draw the free body diagram.

c) A ball having a mass of \(0.5 \mathrm{~kg}\) is moving towards the east with a speed of velocity \(8.0 \mathrm{~ms}^{-1}\). After being hit by a bat it changes its direction and starts moving towards the north with a speed of \(6.0 \mathrm{~ms}^{-1}\). If the time of impact is \(0.1 \mathrm{~s}\), calculate the average force acting on the ball.

d) The vertical circular ride in an amusement park has a radius of \(40 \mathrm{~m}\). You are sitting in a car that is just at the top of the ride. How fast must the car be moving so that you momentarily lift off your seat and feel weightless? Take \(g=10 \mathrm{~ms}^{-2}\).

\section*{PART B}

4. a) A grinding wheel starts from rest and has a constant angular acceleration of \(5 \mathrm{rad} \mathrm{s}^{-2}\). At \(t=5 \mathrm{~s}\) find the total acceleration at a point \(1.0 \mathrm{~m}\) from the axis.

b) An insect of mass \(20 \mathrm{~g}\) crawls from the centre to the outside edge of a rotating disc of mass \(200 \mathrm{~g}\) and radius \(20 \mathrm{~cm}\). The disk was initially rotating at 22.0 rads \(^{-1}\). What will be its final angular velocity? c) The comet Encke has an aphelion distance of \(6.1 \times 10^{11} \mathrm{~m}\) and perihelion distance of \(5.1 \times 10^{11} \mathrm{~m}\). The mass of the sun is \(2.0 \times 10^{30} \mathrm{~kg}\). Calculate the speed of the comet at the perihelion.

d) The mass (in \(\mathrm{kg}\) ) and position coordinates (in \(\mathrm{m}\) ) of a system of three particles \(A\), \(B\) and \(C\) are as follows:

\(\begin{array}{ccc}\text { Particle } & \text { Mass } & \text { Position } \\ A & 2.0 \mathrm{~kg} & (0,0) \\ B & 1.0 \mathrm{~kg} & (2,0) \\ C & 3.0 \mathrm{~kg} & (1,1)\end{array}\)

Calculate the coordinates of the centre of mass of the system.

e) A particle of mass \(10.0 \mathrm{~kg}\), initially moving with a velocity of \(5.0 \mathrm{~ms}^{-1}\) collides elastically with a particle of mass \(5.0 \mathrm{~kg}\), initially moving with a velocity of \(-8.0 \mathrm{~ms}^{-1}\). What are the velocities of the two particles before and after the collision in the centre of mass frame of reference?

5. a) The amplitude of oscillation of a simple harmonic oscillator is \(40 \mathrm{~cm}\). Show that its instantaneous kinetic energy is less than its average kinetic energy when the displacement is \(30 \mathrm{~cm}\).

b) Two collinear harmonic oscillations are represented by:

\[
x_{1}=6 \sin \left(10 \pi t+\frac{\pi}{6}\right) \mathrm{cm} ; x_{2}=8 \sin \left(10 \pi t+\frac{\pi}{3}\right) \mathrm{cm}
\]

Calculate the amplitude, phase constant, and the period of resultant oscillation obtained on superposing these two collinear oscillations.

c) The quality factor of a sonometer wire is 3000 . The wire vibrates at a frequency of \(250 \mathrm{~Hz}\). Calculate the time in which its amplitude will reduce to half of its initial value.

d) A transverse wave travelling in the positive \(x\)-direction is given by \(y(x, t)=6 \sin (8 t-.05 x) \mathrm{cm}\), where \(x\) is in \(\mathrm{cm}\) and \(t\) is in seconds. Calculate the velocity of the wave and the maximum particle velocity.