156- Three forces $\vec{F}_1$, $\vec{F}_2$, $\vec{F}_3$ make pairwise angles of $120^\circ$ with each other. If the magnitudes of the forces are 5, 10, and 15 newtons respectively, find their resultant. How many newtons is it?
(1) zero (2) $5$ (3) $5\sqrt{3}$ (4) $10$

161- According to the figure below, a horizontal force $F$ is applied to a body. What is the minimum value of $F$ (in terms of the body's weight) so that the body remains stationary on the inclined surface? ($\sin 53^\circ = 0.8$, $g = 10\dfrac{m}{s^2}$)
[Figure: A block of mass $m$ on an inclined plane at $53^\circ$ with a horizontal force $F$ applied, $\mu_s = 1$]
(1) $\dfrac{1}{2}$ (2) $\dfrac{3}{5}$ (3) $\dfrac{4}{5}$ (4) $1$

161. A uniform rope of mass $40\,\text{kg}$, as shown in the figure below, rests against a frictionless wall. If the wall exerts a force of $300\,\text{N}$ on the rope, how many newtons does the surface exert horizontally on the rope?
$$\left(g = 10\,\frac{\text{N}}{\text{kg}}\right)$$
[Figure: A rope leaning against a wall at an angle, with the base on a horizontal surface]

• [(1)] $400$
• [(2)] $500$
• [(3)] $600$
• [(4)] $250\sqrt{3}$

51 -- A body with mass $5\,\text{kg}$ is placed on a horizontal surface and the coefficients of static and kinetic friction between the body and the surface are $0.5$ and $0.4$ respectively. If we apply a horizontal force of $26\,\text{N}$ to the body, find the acceleration of the body and the force the body exerts on the surface in SI units. $\left(g = 10\,\dfrac{\text{m}}{\text{s}^2}\right)$

• [(1)] $10\sqrt{29}$ and $0.2$
• [(2)] $25\sqrt{5}$ and $0.2$
• [(3)] $10\sqrt{29}$ and $1.2$
• [(4)] $25\sqrt{5}$ and $1.2$

Two forces are such that the sum of their magnitudes is 18 N and their resultant is 12 N which is perpendicular to the smaller force. Then the magnitudes of the forces are
(1) $12 \mathrm{~N} , 6 \mathrm{~N}$
(2) $13 \mathrm{~N} , 5 \mathrm{~N}$
(3) $10 \mathrm{~N} , 8 \mathrm{~N}$
(4) $16 \mathrm{~N} , 2 \mathrm{~N}$

When forces $F_1, F_2, F_3$ are acting on a particle of mass $m$ such that $F_2$ and $F_3$ are mutually perpendicular, then the particle remains stationary. If the force $F_1$ is now removed then the acceleration of the particle is
(1) $\mathrm{F}_1 / \mathrm{m}$
(2) $\mathrm{F}_2 \mathrm{~F}_3 / \mathrm{mF}_1$
(3) $\left(F_2 - F_3\right) / m$
(4) $\mathrm{F}_2 / \mathrm{m}$

A block rests on a rough inclined plane making an angle of $30 ^ { \circ }$ with the horizontal. The coefficient of static friction between the block and the plane is 0.8 . If the frictional force on the block is 10 N , the mass of the block (in kg ) is (take $\mathrm { g } = 10 \mathrm {~m} / \mathrm { s } ^ { 2 }$ )
(1) 2.0
(2) 4.0
(3) 1.6
(4) 2.5

A block is kept on a frictionless inclined surface with angle of inclination $\alpha$. The incline is given an acceleration a to keep the block stationary. Then a is equal to
(1) $g/\tan\alpha$
(2) $g\operatorname{cosec}\alpha$
(3) g
(4) $g\tan\alpha$

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

The resultant of two forces P N and 3 N is a force of 7 N . If the direction of 3 N force were reversed, the resultant would be $\sqrt { 19 } \mathrm {~N}$. The value of P is
(1) 5 N
(2) 6 N
(3) 3 N
(4) 4 N

An insect crawls up a hemispherical surface very slowly. The coefficient of friction between the insect and the surface is $1/3$. If the line joining the centre of the hemispherical surface to the insect makes an angle $\alpha$ with the vertical, the maximum possible value of $\alpha$ so that the insect does not slip is given by
(1) $\cot \alpha = 3$
(2) $\sec \alpha = 3$
(3) $\operatorname{cosec} \alpha = 3$
(4) $\cos \alpha = 3$

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.
(1) $\theta = \tan ^ { - 1 } ( \mu ) , F = \frac { \mu W } { \sqrt { 1 + \mu ^ { 2 } } }$
(2) $\theta = \tan ^ { - 1 } \left( \frac { 1 } { \mu } \right) , F = \frac { \mu W } { \sqrt { 1 + \mu ^ { 2 } } }$
(3) $\theta = 0 , F = \mu W$
(4) $\theta = \tan ^ { - 1 } \left( \frac { \mu } { 1 + \mu } \right) , F = \frac { \mu W } { 1 + \mu }$

A body of mass 2 kg slides down with an acceleration of $3 \mathrm {~m} / \mathrm { s } ^ { 2 }$ on a rough inclined plane having a slope of $30 ^ { \circ }$ . The external force required to take the same body up the plane with the same acceleration will be: $\left( \mathrm { g } = 10 \mathrm {~m} / \mathrm { s } ^ { 2 } \right)$
(1) 4 N
(2) 14 N
(3) 6 N
(4) 20 N

Two forces $P$ and $Q$, of magnitude $2F$ and $3F$, respectively, are at an angle $\theta$ with each other. If the force $Q$ is doubled, then their resultant also gets doubled. Then, the angle $\theta$ is:
(1) $120 ^ { \circ }$
(2) $60 ^ { \circ }$
(3) $30 ^ { \circ }$
(4) $90 ^ { \circ }$

A block of mass 10 kg is kept on a rough inclined plane as shown in the figure. A force of $3 N$ is applied on the block. The coefficient of static friction between the plane and the block is 0.6 . What should be the minimum value of force $P$, such that the block does not move downward? (take $g = 10 \mathrm {~ms} ^ { - 2 }$)
(1) 23 N
(2) 25 N
(3) 18 N
(4) 32 N

A mass of 10 kg is suspended by a rope of length 4 m, from the ceiling. A force F is applied horizontally at the mid-point of the rope such that the top half of the rope makes an angle of $45 ^ { \circ }$ with the vertical. Then F equals: (Take $\mathrm { g } = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ and the rope to be massless)
(1) 100 N
(2) 90 N
(3) 70 N
(4) 75 N

The resultant of these forces $\overrightarrow{OP}, \overrightarrow{OQ}, \overrightarrow{OR}, \overrightarrow{OS}$ and $\overrightarrow{OT}$ is approximately $\_\_\_\_$ N. [Take $\sqrt{3} = 1.7, \sqrt{2} = 1.4$ Given $\hat{\mathrm{i}}$ and $\hat{\mathrm{j}}$ unit vectors along $x, y$ axis]
(1) $-1.5\hat{\mathrm{i}} - 15.5\hat{\mathrm{j}}$
(2) $9.25\hat{i} + 5\hat{j}$
(3) $3\hat{i} + 15\hat{j}$
(4) $2.5\hat{\mathrm{i}} - 14.5\hat{\mathrm{j}}$

Two blocks ($m = 0.5 \mathrm {~kg}$ and $M = 4.5 \mathrm {~kg}$) are arranged on a horizontal frictionless table as shown in the figure. The coefficient of static friction between the two blocks is $\frac { 3 } { 7 }$. Then the maximum horizontal force that can be applied on the larger block so that the blocks move together is $N$. (Round off to the Nearest Integer) [Take $g$ as $9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$]

Two forces $\bar { F } _ { 1 }$ and $\bar { F } _ { 2 }$ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between $\vec { F } _ { 1 }$ and $\vec { F } _ { 2 }$ is $\cos ^ { - 1 } \left( \frac { 1 } { n } \right)$. The value of $| n |$ is $\_\_\_\_$.

Q1. A particle moves in $x - y$ plane under the influence of a force $\vec { F }$ such that its linear momentum is $\overrightarrow { \mathrm { p } } ( \mathrm { t } ) = \hat { i } \cos ( \mathrm { kt } ) - \hat { j } \sin ( \mathrm { kt } )$. If k is constant, the angle between $\overrightarrow { \mathrm { F } }$ and $\overrightarrow { \mathrm { p } }$ will be :
(1) $\frac { \pi } { 4 }$
(2) $\frac { \pi } { 6 }$
(3) $\frac { \pi } { 2 }$
(4) $\frac { \pi } { 3 }$

Q3. A body of weight 200 N is suspended from a tree branch through a chain of mass 10 kg . The branch pulls the chain by a force equal to (if $g = 10 \mathrm {~m} / \mathrm { s } ^ { 2 }$ ) :
(1) 100 N
(2) 200 N
(3) 300 N
(4) 150 N

Q3. A 1 kg mass is suspended from the ceiling by a rope of length 4 m . A horizontal force ' $F ^ { \prime }$ ' is applied at the mid point of the rope so that the rope makes an angle of $45 ^ { \circ }$ with respect to the vertical axis as shown in figure. The [Figure] magnitude of $F$ is : (Assume that the system is in equilibrium and $g = 10 \mathrm {~m} / \mathrm { s } ^ { 2 }$ )
(1) 10 N
(2) $\frac { 10 } { \sqrt { 2 } } \mathrm {~N}$
(3) 1 N
(4) $\frac { 1 } { 10 \times \sqrt { 2 } } \mathrm {~N}$

Q4. A wooden block of mass 5 kg rests on a soft horizontal floor. When an iron cylinder of mass 25 kg is placed on the top of the block, the floor yields and the block and the cylinder together go down with an acceleration of $0.1 \mathrm {~ms} ^ { - 2 }$. The action force of the system on the floor is equal to:
(1) 196 N
(2) 291 N
(3) 294 N
(4) 297 N

Q8. A small ball of mass $m$ and density $\rho$ is dropped in a viscous liquid of density $\rho _ { 0 }$. After sometime, the ball falls with constant velocity. The viscous force on the ball is :
(1) $m g \left( 1 - \rho \rho _ { 0 } \right)$
(2) $m g \left( 1 + \frac { \rho } { \rho _ { 0 } } \right)$
(3) $m g \left( \frac { \rho _ { 0 } } { \rho } - 1 \right)$
(4) $m g \left( 1 - \frac { \rho _ { 0 } } { \rho } \right) _ { \nabla }$