Constrained Motion : General Approach

Q.

If $\mathrm{t}=\frac{{\mathrm{v}}^2}{2}$, then $\left(-\frac{\mathrm{df}}{\mathrm{dt}}\right)$ is equal to, (where $\mathrm{f}$ is acceleration)

Q. In the system shown below, $M_A=3\text{kg},M_B=4\text{kg}$ and $M_C=8\text{kg}$. Co-efficient of friction between any two surfaces is 0.25. Pulley is frictionless and string is massless. If $A$ is connected to the wall through a massless rigid rod, then choose the correct option(s)

1. Value of $F$ to keep $C$ moving with constant speed is $80\text{N}$
2. Value of $F$ to keep $C$ moving with constant speed is $120\text{N}$
3. If $F$ is $200\text{N}$, then acceleration of $B$ is $10\text{m}/{\text{s}}^2$
4. To slide $C$ towards left, $F$ should be at least $50\text{N}$

Q. Assuming the string to be inextensible and surface and pulley to be frictionless, velocity of block $B$ $\left(V_B\right)$ in the given figure is

1. $2\text{m/s}$
2. $6\text{m/s}$
3. $8\text{m/s}$
4. $10\text{m/s}$

Q. Find the velocity of block $B$ i.e. $v_B$ in the shown figure.

1. $2v$, downward
2. $2v$, upward
3. $\frac{v}{3}$, upward
4. $\frac{v}{2}$, upward

Q. Find the tension in the string connected to the block of mass $4\text{kg}$. Assume the strings and pulleys to be ideal. (Take $g=10m/s^2)$

1. $\frac{100}{9}N$
2. $\frac{40}{3}N$
3. $\frac{15}{2}N$
4. $\frac{50}{3}N$

Q.

The pull $\mathrm{P}$ is just sufficient to keep the block of weight $14\mathrm{N}$ in equilibrium as shown. Pulleys are ideal. Find the value of $\mathrm{P}$.

1. $4\mathrm{N}$

2. $6\mathrm{N}$

3. $8\mathrm{N}$

4. $2\mathrm{N}$

Q. In the given constraint, if the string is inextensible and pulley is frictionless, then magnitude of velocity of block $A$ $\left(V_A\right)$ is

1. $10\text{m/s}$
2. $20\text{m/s}$
3. $15\text{m/s}$
4. $5\text{m/s}$

Q.

The x and y coordinates of a particle at any time t are given by

x = 7t + 4t² and y = 5t,

Where x and y are in metres and t in seconds. The acceleration of a particle at t = 5s is.

1. Zero

2. 8 m/s²

3. 20 m/s²

4. 40 m/s²

Q. In the figure shown below, acceleration of block $A$ is $1{\text{m/s}}^2$ upwards, acceleration of block $B$ is $7{\text{m/s}}^2$ upwards and acceleration of block $C$ is $2{\text{m/s}}^2$ upwards. Then what will be the acceleration of block $D$? Assume that the pulleys are frictionless and strings are inextensible.

1. $7{\text{m/s}}^2$ upwards
2. $2{\text{m/s}}^2$ downwards
3. $10{\text{m/s}}^2$ downwards
4. $8{\text{m/s}}^2$ upwards

Q. As shown in figure, all the surfaces are frictionless and the pulleys and the string are light. The magnitude of acceleration of block $2M$ is

1. $\frac{g}{3}$
2. $\frac{g}{6}$
3. $\frac{2g}{3}$
4. $\frac{g}{4}$

Q. Find velocity of block $B$ at the instant shown in figure. Assume the surface and pulleys to be frictionless and string to be inextensible.

1. $25\text{m/s}$
2. $20\text{m/s}$
3. $22\text{m/s}$
4. $30\text{m/s}$

Q. The pulley and strings shown in the figure below are massless. Find the Acceleration of system.
(Take $g=10{\text{m/s}}^2$)

1. $2{\text{m/s}}^2$
2. $1{\text{m/s}}^2$
3. $1.5{\text{m/s}}^2$
4. $2.5{\text{m/s}}^2$

Q. Find the acceleration of block $C$ i.e. $a_C$ in the shown figure.

1. $a_C=6{\text{m/s}}^2$, upwards
2. $a_C=4{\text{m/s}}^2$, upwards
3. $a_C=5{\text{m/s}}^2$, upwards
4. $a_C=2{\text{m/s}}^2$, upwards

Q. The string shown in the figure is passing over a small smooth pulley rigidly attached to trolley A. If the speed of the trolley is constant and equal to $v_A$, speed and acceleration of block B at the instant is $v_B$ and $a_B$ respectively, which of the following option(s) is/are correct?

1. $v_B=v_A,a_B=0$
2. $a_B=0$
3. $v_B=\frac{3}{5}{v}_A$
4. $a_B=\frac{16v_A^2}{125}$

Q. A rod of mass $2\text{m}$ moves vertically downward on the surface of wedge of mass as shown in the figure. Find the relation between velocity of rod and that of the wedge at any instant.

1. $u=v\tan \theta$
2. $u=v\tan 2\theta$
3. $v=u\tan \theta$
4. $v=u\sin \theta$

Q. Figure shows two blocks A and B connected to an ideal pulley string system. In this system when bodies are released then: (neglect friction and take $g=10\frac{m}{s^2})$

1. Acceleration of block A is 1 $\frac{m}{s^2}$
2. Acceleration of block A is 2 $\frac{m}{s^2}$
3. Tension in string connected to block B is 40 N
4. Tension in string connected to block B is 80 N

Q. In the figure shown, find the acceleration (in $m/s^2$) of the $2kg$ body and the tension (in $N$) in the string. (String is massless and inextensible)

1. $0,2g$
2. $g,2g$
3. $2g,g$
4. $0,g$

Q. Assuming the string to be inextensible and pulley and surface to be frictionless, the speed of block B in the pulley block system as shown in figure is

1. $5m/s$
2. $6.5m/s$
3. $7.5m/s$
4. $10m/s$

Q.

At a given instant, block $A$ is moving with velocity of $5\text{m/s}$ upwards. What is the velocity of block $B$ at that time?

1. $15\text{m/s}$ (Downward)
2. $15\text{m/s}$ (Upward)
3. $10\text{m/s}$ (Downward)
4. $10\text{m/s}$ (Upward)

Q. Blocks B and C are connected by a single inextensible cable, with this cable being wrapped around pulleys at D and E. In addition, the cable is wrapped around a pulley attached to block A as shown. Assume the radii of the pulleys to be small. Blocks B and C move downward with speeds of $V_B=6ft/s$ and $V_C=18ft/s,$ respectively. Determine the velocity of block A when $S_A=4ft.$

1. $6ft/s$
2. $15ft/s$
3. $18ft/s$
4. $20ft/s$

Q. In the figure shown, all pulleys are massless and the strings are light and inextensible. What is the acceleration of the pulley $P_4$? (All surfaces are smooth and take $g=10m/s^2$.)

1. $2.25{\text{m/s}}^2$ towards left
2. $2.25{\text{m/s}}^2$ towards right
3. $9{\text{m/s}}^2$ towards left
4. $9{\text{m/s}}^2$ towards right

Q. If acceleration of block $A$ is $2{\text{m/s}}^2$ to left and acceleration of block $B$ is $1{\text{m/s}}^2$ to left, then what will be the acceleration of block $C$? Assume that the surface and pulleys are smooth and string is inextensible.

1. $1{\text{m/s}}^2$ upwards
2. $1{\text{m/s}}^2$ downwards
3. $2{\text{m/s}}^2$ downwards
4. $2{\text{m/s}}^2$ upwards

Q. In the arrangement shown in the figure, if $v_1$ and $v_2$ are instantaneous velocities of masses $M_1$ and $M_2$ respectively, and angle $ACB=2\theta$ at the instant, then

1. $\theta ={\cos }^{-1}\left(\frac{v_2}{2v_1}\right)$
2. $\theta ={\cos }^{-1}\left(\frac{v_1}{2v_2}\right)$
3. $\theta ={\tan }^{-1}\left(\frac{v_1}{2v_2}\right)$
4. $\theta ={\sin }^{-1}\left(\frac{v_1}{v_2}\right)$

Q. Find the realation between the velocity of block $A$ and velocity of block $B$ in the shown figure.

1. $8v_A+2v_B=4v$
2. $4v_A+v_B=2v$
3. $8v_A+v_B=6v$
4. $8v_A+3v_B=2v$

Q. Find the acceleration of block $B$ in the shown figure.

1. $2{\text{m/s}}^2$, leftward
2. $\frac{4}{3}{\text{m/s}}^2$, leftward
3. $\frac{2}{3}{\text{m/s}}^2$, leftward
4. $3{\text{m/s}}^2$, rightward

Q. Three blocks 1, 2 and 3 are arranged as shown in the figure. The velocities of the blocks $v_1,v_2$ and $v_3$ are shown in the figure. What is the relationship between $v_1,v_2$ and $v_3$?

1. $2v_1+v_2=v_3$
2. $v_1+v_2=v_3$
3. $v_1+2v_2=v_3$
4. $2v_1+v_2=2v_3$

Q. Find the acceleration of block of mass $m$ along the inclined plane $\left(a_x\right)$ if the acceleration of wedge, $b=4{\text{m/s}}^2$. All accelerations are w.r.t. to the ground and assume that the surface and pulley are smooth and the string is inextensible.

1. $2{\text{m/s}}^2$
2. $3{\text{m/s}}^2$
3. $4{\text{m/s}}^2$
4. $1{\text{m/s}}^2$

Q. Three blocks are connected by different inextensible string as shown in the figure. If the velocity of block $A$ is $10\text{m/s}$, the velocity of block $C$ is

1. $20\text{m/s}$
2. $5\text{m/s}$
3. $10\text{m/s}$
4. $8\text{m/s}$

Q. In the given constraint, if the the surface and the pulley are frictionless and the string is inextensible, then velocity of the block $A$parallel to the smooth surface is

1. $\frac{5}{\sqrt{3}}\text{m/s}$
2. $3\text{m/s}$
3. $5\text{m/s}$
4. $10\text{m/s}$

Q. Find the velocity of the hanging block if the velocities of the free ends of the rope are as indicated in the figure. Assume that the pulleys are frictionless and string is inextensible.

1. $3/2m/s\uparrow$
2. $3/2m/s\downarrow$
3. $1/2m/s\uparrow$
4. $1/2m/s\downarrow$