Mechanics PYQ — Page 78

A body of mass ' $m$ ' is tied to one end of a spring and whirled round in a horizontal plane with a constant angular velocity. The elongation in the spring is $1 \mathrm{~cm}$. If the angular velocity is doubled, the elongation in the spring is $5 \mathrm{~cm}$. The original length of the spring is :

A body starts from rest on a long inclined plane of slope $45^{\circ}$. The coefficient of friction between the body and the plane varies as $\mu=0.3 x$, where $x$ is distance travelled down the plane. The body will have maximum speed (for $g=10 \mathrm{~m} / \mathrm{s}^2$ ) when $x=$

A projectile of mass $M$ is fired so that the horizontal range is $4 \mathrm{~km}$. At the highest point the projectile explodes in two parts of masses $M / 4$ and $3 M / 4$ respectively and the heavier part starts falling down vertically with zero initial speed. The horizontal range (distance from point of firing) of the lighter part is :

Two springs of force constants $300 \mathrm{~N} / \mathrm{m}$ (Spring A) and $400 \mathrm{~N} / \mathrm{m}$ (Spring B) are joined together in series. The combination is compressed by $8.75 \mathrm{~cm}$. The ratio of energy stored in $\mathrm{A}$ and $\mathrm{B}$ is $\frac{E_A}{E_B}$. Then $\frac{E_A}{E_B}$ is equal to:

A particle of mass $2 \mathrm{~kg}$ is moving such that at time $t$, its position, in meter, is given by $\vec{r}(t)=5 \hat{i}-2 t^2 \hat{j}$. The angular momentum of the particle at $t=2 s$ about the origin in $\mathrm{kg} \mathrm{m}^{-2} \mathrm{~s}^{-1}$ is :

What is the minimum energy required to launch a satellite of mass $\text{m}$ from the surface of a planet of mass $\text{M}$ and radius $\text{R}$ in a circular orbit at an altitude of $\text{2R}$?

Air of density $1.2 \mathrm{~kg} \mathrm{~m}^{-3}$ is blowing across the horizontal wings of an aeroplane in such a way that its speeds above and below the wings are $150 \mathrm{~ms}^{-1}$ and $100 \mathrm{~ms}^{-1}$, respectively. The pressure difference between the upper and lower sides of the wings, is :

The distance travelled by a body moving along a line in time $t$ is proportional to $t^3$. The acceleration-time $(a, t)$ graph for the motion of the body will be

In a cylindrical water tank, there are two small holes $A$ and $B$ on the wall at a depth of $h_1$, from the surface of water and at a height of $h_2$ from the bottom of water tank. Surface of water is at height of $h_2$ from the bottom of water tank. Surface of water is at heigh $H$ from the bottom of water tank. Water coming out from both holes strikes the ground at the same point $S$. Find the ratio of $h_1$ and $h_2$ 

A block of weight $W$ rests on a horizontal floor with coefficient of static friction $\mu$. It is desired to make the block move by applying minimum amount of force. The angle $\theta$ from the horizontal at which the force should be applied and magnitude of the force $F$ are respectively.

A moving particle of mass $m$, makes a head on elastic collision with another particle of mass $2 m$, which is initially at rest. The percentage loss in energy of the colliding particle on collision, is close to

A boy can throw a stone up to a maximum height of $10 \mathrm{~m}$. The maximum horizontal distance that the boy can throw the same stone up to will be

A spectrometer gives the following reading when used to measure the angle of a prism. Main scale reading: $58.5$ degree Vernier scale reading : $09$ divisions Given that $1$ division on main scale corresponds to $0.5$ degree. Total divisions on the vernier scale is $30$ and match with $29$ divisions of the main scale. The angle of the prism from the above data

A student measured the diameter of a wire using a screw gauge with the least count $0.001 \mathrm{~cm}$ and listed the measurements. The measured value should be recorded as

The amount of heat produced in an electric circuit depends upon the current $(I)$, resistance $(R)$ and time $(t)$. If the error made in the measurements of the above quantities are $2 \%, 1 \%$ and $1 \%$ respectively then the maximum possible error in the total heat produced will be

A spring is compressed between two blocks of masses $m_1$ and $m_2$ placed on a horizontal frictionless surface as shown in the figure. When the blocks are released, they have initial velocity of $v_1$ and $v_2$ as shown. The blocks travel distances $x_1$ and $x_2$ respectively before coming to rest. The ratio $\left(\frac{x_1}{x_2}\right)$ is 

A circular hole of diameter $\mathrm{R}$ is cut from a disc of mass $M$ and radius $R$; the circumference of the cut passes through the centre of the disc. The moment of inertia of the remaining portion of the disc about an axis perpendicular to the disc and passing through its centre is

This question has Statement 1 and Statement 2. Of the four choices given after the Statements, choose the one that best describes the two Statements. Statement 1: When moment of inertia $I$ of a body rotating about an axis with angular speed $\omega$ increases, its angular momentum $L$ is unchanged but the kinetic energy $K$ increases if there is no torque applied on it. Statement 2: $L=I \omega$, kinetic energy of rotation $=\frac{1}{2} I \omega^2$

Which graph correctly presents the variation of acceleration due to gravity with the distance from the centre of the earth (radius of the earth $=R_E$ )?

The load versus elongation graphs for four wires of same length and made of the same material are shown in the figure. The thinnest wire is represented by the line 

An engine pumps water continuously through a hose. Water leaves the hose with velocity $v$ and $m$ is mass per unit length of the water jet. If this jet hits a surface and came to rest instantaneously, the force on the surface is

A ball is dropped vertically downwards from a height $h$ above the ground. It hits the ground inelastically and bounces up vertically. Neglecting subsequent motion and air resistance, which of the following graph represents variation between speed $(v)$ and height $(h)$ correctly?

The graph of an object's motion (along the $x-$ axis) is shown in the figure. The instantaneous velocity of the object at points $A$ and $B$ are $v_A$ and $v_B$ respectively. Then 

Two point masses of mass $m_1=f M$ and $m_2=(1-f) M(f < 1)$ are in outer space (far from gravitational influence of other objects) at a distance $R$ from each other. They move in circular orbits about their centre of mass with angular velocities $\omega_1$ for $m_1$ and $\omega_2$ for $m_2$. In that case