Mechanics PYQ — Page 8

A uniform circular disc of radius ' R ' and mass ' M ' is rotating about an axis perpendicular to its plane and passing through its centre. A small circular part of radius $\mathrm{R} / 2$ is removed from the original disc as shown in the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above. 

A wheel is rolling on a plane surface. The speed of a particle on the highest point of the rim is $8 \mathrm{~m} / \mathrm{s}$. The speed of the particle on the rim of the wheel at the same level as the centre of wheel, will be :

 A wheel of radius 0.2 m rotates freely about its center when a string that is wrapped over its rim is pulled by force of 10 N as shown in figure. The established torque produces an angular acceleration of $2 \mathrm{~rad} / \mathrm{s}^2$. Moment of inertia of the wheel is _____ $\mathrm{kg} \mathrm{m}^2$. (Acceleration due to gravity $=10 \mathrm{~m} / \mathrm{s}^2$)

Match List - I with List - II. 

If a satellite orbiting the Earth is 9 times closer to the Earth than the Moon, what is the time period of rotation of the satellite? Given rotational time period of Moon $=27$ days and gravitational attraction between the satellite and the moon is neglected.

The least count of a screw guage is 0.01 mm . If the pitch is increased by $75 \%$ and number of divisions on the circular scale is reduced by $50 \%$, the new least count will be _____ $\times 10^{-3} \mathrm{~mm}$

A capillary tube of radius 0.1 mm is partly dipped in water (surface tension $70 \mathrm{dyn} / \mathrm{cm}$ and glass water contact angle $\simeq 0^{\circ}$) with $30^{\circ}$ inclined with vertical. The length of water risen in the capillary is ________ cm. $\left(\right.$ Take $\left.\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^2\right)$

Two liquids A and B have $\theta_{\mathrm{A}}$ and $\theta_{\mathrm{B}}$ as contact angles in a capillary tube. If $K=\cos \theta_A / \cos \theta_B$. then identify the correct statement:

Match List-I with List-II. $\begin{array}{|l|l|l|l|} \hline & \text{List-I} & & \text{List-II} \\ \hline \text{(A)} & \text{Mass density} & \text{(I)} & {\left[\mathrm{ML}^2 \mathrm{~T}^{-3}\right]} \\ \hline \text{(B)} & \text{Impulse} & \text{(II)} & {\left[\mathrm{MLT}^{-1}\right]} \\ \hline \text{(C)} & \text{Power} & \text{(III)} & {\left[\mathrm{ML}^2 \mathrm{~T}^0\right]} \\ \hline \text{(D)} & \text{Moment of inertia} & \text{(IV)} & {\left[\mathrm{ML}^{-3} \mathrm{~T}^0\right]} \\ \hline\end{array}$ Choose the correct answer from the options given below :

A bob of mass $m$ is suspended at a point $O$ by a light string of length $l$ and left to perform vertical motion (circular) as shown in figure. Initially, by applying horizontal velocity $v_0$ at the point ' A ', the string becomes slack when, the bob reaches at the point ' D '. The ratio of the kinetic energy of the bob at the points $B$ and $C$ is ______ -. 

Moment of inertia of a rod of mass ' M ' and length 'L' about an axis passing through its center and normal to its length is ' $\alpha$ '. Now the rod is cut into two equal parts and these parts are joined symmetrically to form a cross shape. Moment of inertia of cross about an axis passing through its center and normal to plane containing cross is :

An object of mass ' m ' is projected from origin in a vertical $x y$ plane at an angle $45^{\circ}$ with the x axis with an initial velocity $\mathrm{v}_0$. The magnitude and direction of the angular momentum of the object with respect to origin, when it reaches at the maximum height, will be [ g is acceleration due to gravity]

 A body of mass 1 kg is suspended with the help of two strings making angles as shown in figure. Magnitude of tensions $T_1$ and $T_2$, respectively, are (in N) :

Water flows in a horizontal pipe whose one end is closed with a valve. The reading of the pressure gauge attached to the pipe is $P_1$. The reading of the pressure gauge falls to $P_2$ when the valve is opened. The speed of water flowing in the pipe is proportional to

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) :  Three identical spheres of same mass undergo one dimensional motion as shown in figure with initial velocities $v_{\mathrm{A}}=5 \mathrm{~m} / \mathrm{s}, v_{\mathrm{B}}=2 \mathrm{~m} / \mathrm{s}, v_{\mathrm{C}}=4 \mathrm{~m} / \mathrm{s}$. If we wait sufficiently long for elastic collision to happen, then $v_{\mathrm{A}}=4 \mathrm{~m} / \mathrm{s}, v_{\mathrm{B}}=2 \mathrm{~m} / \mathrm{s}$, $v_{\mathrm{C}}=5 \mathrm{~m} / \mathrm{s}$ will be the final velocities. Reason (R): In an elastic collision between identical masses, two objects exchange their velocities. In the light of the above statements, choose the correct answer from the options given below :

Match List - I with List - II. $$ \begin{array}{llll} & \text { List - I } & & \text { List - II } \\ \text { (A) } & \text { Permeability of free space } & \text { (I) }\left[\mathrm{M} \mathrm{~L}^2 \mathrm{~T}^{-2}\right] \\ \text { (B) } & \text { Magnetic field } & \text { (II) }\left[\mathrm{M} \mathrm{~T}^{-2} \mathrm{~A}^{-1}\right] \\ \text { (C) } & \text { Magnetic moment } & \text { (III) }\left[\mathrm{M} \mathrm{~L} \mathrm{~T}^{-2} \mathrm{~A}^{-2}\right] \\ \text { (D) } & \text { Torsional constant } & \text { (IV) }\left[\mathrm{L}^2 \mathrm{~A}\right] \end{array} $$ Choose the correct answer from the options given below :

Two projectiles are fired from ground with same initial speeds from same point at angles $\left(45^{\circ}+\alpha\right)$ and $\left(45^{\circ}-\alpha\right)$ with horizontal direction. The ratio of their times of flights is

Match List-I with List-II. $\begin{array}{ll} \text{List-I} & \text{List-II} \\ \text{(A) Coefficient of viscosity} & \text{(I) } \left[\mathrm{ML}^0 \mathrm{~T}^{-3}\right] \\ \text{(B) Intensity of wave} & \text{(II) } \left[\mathrm{ML}^{-2} \mathrm{~T}^{-2}\right] \\ \text{(C) Pressure gradient} & \text{(III) } \left[\mathrm{M}^{-1} \mathrm{LT}^2\right] \\ \text{(D) Compressibility} & \text{(IV) } \left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right]\end{array}$ Choose the correct answer from the options given below :

Which one of the following forces cannot be expressed in terms of potential energy?

A sand dropper drops sand of mass $m(t)$ on a conveyer belt at a rate proportional to the square root of speed $(v)$ of the belt, i.e. $\frac{\mathrm{dm}}{\mathrm{dt}} \propto \sqrt{v}$. If P is the power delivered to run the belt at constant speed then which of the following relationship is true?

A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is $\sqrt{\frac{x}{5}}$ where $\mathrm{x}=$ _____

A solid sphere with uniform density and radius R is rotating initially with constant angular velocity $\left(\omega_1\right)$ about its diameter. After some time during the rotation its starts loosing mass at a uniform rate, with no change in its shape. The angular velocity of the sphere when its radius become $R / 2$ is $x \omega_1$. The value of $x$ is ________

The torque due to the force $(2 \hat{i}+\hat{j}+2 \hat{k})$ about the origin, acting on a particle whose position vector is $(\hat{i}+\hat{j}+\hat{k})$, would be

Given below are two statements: one is labelled as Assertion $\mathbf{A}$ and the other is labelled as Reason $\mathbf{R}$ Assertion A: In a central force field, the work done is independent of the path chosen. Reason R: Every force encountered in mechanics does not have an associated potential energy. In the light of the above statements, choose the most appropriate answer from the options given below