Laws of Motion

A wire, which passes through the hole in a small bead, is bent in the form of quarter of a circle. The wire is fixed vertically on ground as shown in the figure. The bead is released from near the top of the wire and it slides along the wire without friction. As the bead moves from A to B, the force it applies on the wire is

A block of mass $m_1=1$ kg another mass $m_2=2$ kg, are placed together (see figure) on an inclined plane with angle of inclination $\theta$. Various values of $\theta$ are given in List I. The coefficient of friction between the block $m_1$ and the plane is always zero. The coefficient of static and dynamic friction between the block $m_2$ and the plane are equal to $\mu=0.3$. In List II expressions for the friction on block $m_2$ are given. Match the correct expression of the friction in List II with the angles given in List I, and choose the correct option. The acceleration due to gravity is denoted by $g$.

[Useful information: $\tan(5.5^\circ)\approx0.1$; $\tan(11.5^\circ)\approx0.2$; $\tan(16.5^\circ)\approx0.3$]

A person in a lift is holding a water jar, which has a small hole at the lower end of its side. When the lift is at rest, the water jet coming out of the hole hits the floor of the lift at a distance of 1.2 m from the person. In the following, state of the lift's motion is given in List I and the distance where the water jet hits the floor of the lift is given in List II. Match the statements from List I with those in List II and select the correct answer using the code given below the list.

A block of mass m is placed on a surface with a vertical cross section given by $y=\dfrac{x^3}{6}$. If the coefficient of friction is 0.5, the maximum height above the ground at which the block can be placed without slipping is :

A bob of mass m attached to an inextensible string of length $l$ is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed $\omega$ rad/s about the vertical. About the point of suspension :

Four particles, each of mass M and equidistant from each other, move along a circle of radius R under the action of their mutual gravitational attraction. The speed of each particle is :

Water is flowing at a speed of 1.5 ms$^{-1}$ through a horizontal tube of cross-sectional area $10^{-2}$ m$^2$ and you are trying to stop the flow by your palm. Assuming that the water stops immediately after hitting the palm, the minimum force that you must exert should be (density of water $=10^3$ kgm$^{-3}$).

A block A of mass 4 kg is placed on another block B of mass 5 kg, and the block B rests on a smooth horizontal table. If the minimum force that can be applied on A so that both the blocks move together is 12 N, the maximum force that can be applied on B for the blocks to move together will be :

A body of mass 5 kg under the action of constant force $\vec F=F_x\hat i+F_y\hat j$ has velocity at $t=0$ s as $\vec v=\left(6\hat i-2\hat j\right)$ m/s and at $t=10$ s as $\vec v=+6\hat j$ m/s. The force $\vec F$ is :

A small ball of mass m starts at a point A with speed $v_o$ and moves along a frictionless track AB as shown. The track BC has coefficient of friction $\mu$. The ball comes to stop at C after travelling a distance L which is :

A 4 g bullet is fired horizontally with a speed of 300 m/s into 0.8 kg block of wood at rest on a table. If the coefficient of friction between the block and the table is 0.3, how far will the block slide approximately ?

A heavy box is to be dragged along a rough horizontal floor. To do so, person A pushes it at an angle 30° from the horizontal and requires a minimum force $F_A$, while person B pulls the box at an angle 60° from the horizontal and needs minimum force $F_B$. If the coefficient of friction between the box and the floor is $\dfrac{\sqrt3}5$, the ratio $\dfrac{F_A}{F_B}$ is :

The magnitude of the normal reaction that acts on the block at the point $Q$ is

A small mass $m$ is attached to a massless string whose other end is fixed at $P$ as shown in the figure. The mass is undergoing circular motion in the $x$-$y$ plane with centre at $O$ and constant angular speed $\omega$. If the angular momentum of the system, calculated about $O$ and $P$ are denoted by $\vec L_O$ and $\vec L_P$ respectively, then

A small block of mass of 0.1 kg lies on a fixed inclined plane $PQ$ which makes an angle $\theta$ with the horizontal. A horizontal force of 1 N acts on the block through its center of mass as shown in the figure. The block remains stationary if (take $g=10\ \mathrm{m/s^2}$)

A ball of mass (m) 0.5 kg is attached to the end of a string having length (L) 0.5 m. The ball is rotated on a horizontal circular path about vertical axis. The maximum tension that the string can bear is 324 N. The maximum possible value of angular velocity of ball (in radian/s) is

A boy is pushing a ring of mass 2 kg and radius 0.5 m with a stick as shown in the figure. The stick applies a force of 2 N on the ring and rolls it without slipping with an acceleration of 0.3 m/s². The coefficient of friction between the ground and the ring is large enough that rolling always occurs and the coefficient of friction between the stick and the ring is (P/10). The value of P is

A block is moving on an inclined plane making an angle 45° with the horizontal and the coefficient of friction is $\mu$. The force required to just push it up the inclined plane is 3 times the force required to just prevent it from sliding down. If we define $N=10\mu$, then $N$ is

A train is moving along a straight line with a constant acceleration '$a$'. A boy standing in the train throws a ball forward with a speed of 10 m/s, at an angle of 60° to the horizontal. The boy has to move forward by 1.15 m inside the train to catch the ball back at the initial height. The acceleration of the train, in m/s², is

A block of mass 0.18 kg is attached to a spring of force-constant 2 N/m. The coefficient of friction between the block and the floor is 0.1. Initially the block is at rest and the spring is un-stretched. An impulse is given to the block as shown in the figure. The block slides a distance of 0.06 m and comes to rest for the first time. The initial velocity of the block in m/s is $V=N/10$. Then $N$ is