 # IGNOU BPHCT-131 Solved Assignment 2023 | B.Sc (G) CBCS

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

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Details For BPHCT-131 Solved Assignment

## IGNOU BPHCT-131 Assignment Question Paper 2023

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.

$$Sin^2\left(\theta \:\right)+Cos^2\left(\theta \right)=1$$

## BPHCT-131 Sample Solution 2023

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$$2\:cos\:\theta \:sin\:\phi =sin\:\left(\theta +\phi \right)-sin\:\left(\theta -\phi \right)$$

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