A particle located at $x = 0$ at time $t = 0$, starts moving along the positive $x$-direction with a velocity '$v$' that varies as $v = \alpha \sqrt{x}$. The displacement of the particle varies with time as (1) $t^{3}$ (2) $t^{2}$ (3) $t$ (4) $t^{1/2}$
A body falling from rest under gravity passes a certain point $P$. It was at a distance of 400 m from $P$, $4$ s prior to passing through $P$. If $g = 10$ m/s$^2$, then the height above the point P from where the body began to fall is (1) 720 m (2) 900 m (3) 320 m (4) 680 m
A mass of M kg is suspended by a weightless string. The horizontal force that is required to displace it until the string makes an angle of $45^{\circ}$ with the initial vertical direction is (1) $Mg(\sqrt{2} - 1)$ (2) $Mg(\sqrt{2} + 1)$ (3) $Mg\sqrt{2}$ (4) $\frac{Mg}{\sqrt{2}}$
A player caught a cricket ball of mass 150 g moving at a rate of $20$ m/s. If the catching process is completed in 0.1 s, the force of the blow exerted by the ball on the hand of the player is equal to (1) 300 N (2) 150 N (3) 3 N (4) 30 N
A particle of mass 100 g is thrown vertically upwards with a speed of $5$ m/s. The work done by the force of gravity during the time the particle goes up is (1) 0.5 J (2) $-0.5$ J (3) $-1.25$ J (4) 1.25 J
A ball of mass 0.2 kg is thrown vertically upwards by applying a force by hand. If the hand moves 0.2 m while applying the force and the ball goes upto 2 m height further, find the magnitude of the force. Consider $g = 10$ m/s$^2$ (1) 22 N (2) 4 N (3) 16 N (4) 20 N
The potential energy of a 1 kg particle free to move along the $x$-axis is given by $$V(x) = \left(\frac{x^4}{4} - \frac{x^2}{2}\right) \text{J}$$ The total mechanical energy of the particle is 2 J. Then, the maximum speed (in m/s) is (1) 2 (2) $3/\sqrt{2}$ (3) $\sqrt{2}$ (4) $1/\sqrt{2}$
A bomb of mass 16 kg at rest explodes into two pieces of masses of 4 kg and 12 kg. The velocity of the 12 kg mass is $4$ ms$^{-1}$. The kinetic energy of the other mass is (1) 96 J (2) 144 J (3) 288 J (4) 192 J
Consider a two particle system with particles having masses $m_1$ and $m_2$. If the first particle is pushed towards the centre of mass through a distance $d$, by what distance should the second particle be moved, so as to keep the centre of mass at the same position? (1) $d$ (2) $\frac{m_2}{m_1}$ d (3) $\frac{m_1}{m_1 + m_2} d$ (4) $\frac{m_1}{m_2} d$
A force of $-F\hat{k}$ acts on $O$, the origin of the coordinate system. The torque about the point $(1, -1)$ is (1) $-F(\hat{i} - \hat{j})$ (2) $F(\hat{i} - \hat{j})$ (3) $-F(\hat{i} + \hat{j})$ (4) $F(\hat{i} + \hat{j})$
A thin circular ring of mass $m$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Two objects each of mass $M$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity $\omega' =$ (1) $\frac{\omega m}{(m + 2M)}$ (2) $\frac{\omega(m + 2M)}{m}$ (3) $\frac{\omega(m - 2M)}{(m + 2M)}$ (4) $\frac{\omega m}{(m + M)}$
Four point masses, each of value $m$, are placed at the corners of a square $ABCD$ of side $\ell$. The moment of inertia through $A$ and parallel to $BD$ is (1) $m\ell^2$ (2) $2m\ell^2$ (3) $3m\ell^2$ (4) $3m\ell^2$
Starting from the origin, a body oscillates simple harmonically with a period of 2 s. After what time will its kinetic energy be $75\%$ of the total energy? (1) $\frac{1}{12}$ s (2) $\frac{1}{6}$ s (3) $\frac{1}{4}$ s (4) $\frac{1}{3}$ s
The maximum velocity of a particle, executing simple harmonic motion with an amplitude 7 mm, is $4.4$ m/s. The period of oscillation is (1) 100 s (2) 0.01 s (3) 10 s (4) 0.1 s
A coin is placed on a horizontal platform which undergoes vertical simple harmonic motion of angular frequency $\omega$. The amplitude of oscillation is gradually increased. The coin will leave contact with the platform for the first time (1) at the highest position of the platform (2) at the mean position of the platform (3) for an amplitude of $\frac{g}{\omega^2}$ (4) for an amplitude of $\frac{g^2}{\omega^2}$