Constrained Motion : General Approach

Q.

Water in a bucket is whirled in a vertical circle with a string attached to it. The water does not fall down even when the bucket is inverted at the top of its path. We conclude that in this position.

1. mg =

2. mg is greater than

3. mg is not greater than

4. mg is not less than

Q.

A car travels half the distance with constant velocity of $40kmph$ and the remaining half with a constant velocity of $60kmph$. The average velocity of the car in $kmph$ is

1. $40$

2. $45$

3. $48$

4. $50$

Q.

The force required to move a body up a rough inclined plane is double the force required to prevent the body from sliding down the plane. The coefficient of friction, when the angle of inclination of the plane is $60°$ is

1. $\frac{1}{3}$

2. $\frac{1}{\sqrt{2}}$

3. $\frac{1}{\sqrt{3}}$

4. $\frac{1}{2}$

Q.

Two blocks each of mass M are connected to the ends of a light frame as shown in below figure. The frame is rotated about the vertical line of symmetry. The rod breaks if the tension in it exceeds $T_0$. Find the maximum frequency with which the frame may be rotated without breaking the rod.

1. 

2. 

3. 

4. 

Q. A pendulum in vertical plane having length of stringI and mass of bob m is kept horizontal. Now the bob-is released from horizontal position. The tension instring when it makes an angle of 60^° with vertical

Q. A train has to negotiate a curve of radius $2000\text{m}$. By how much should be the outer rail be raised with respect to the inner rail for a safe travelling speed of $72{\text{kmh}}^{-1}$?
The distance between the rails is $1\text{m}$.
$($Take $g=10{\text{m/s}}^2)$

1. $1\text{cm}$
2. $4.5\text{m}$
3. $3\text{cm}$
4. $2\text{cm}$

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. simple pendulum of length 1 m having a bob of mass 1 kg is hanging from a rigid support. If the bob is projected horizontally with a velocity V35 m/s, then the tension in string is 6K newtons when angle made by the string from vertical is 60^° The value of K is

Q. System is shown in the figure and man is pulling the rope from both sides with constant speed $‘u^{\prime }$. Then what will be the velocity of the block, if the rope is inextensible?

1. $\frac{3u}{4}$
2. $\frac{u}{2}$
3. $\frac{u}{4}$
4. $u$

Q. In the arrangement shown in the figure, accelerations of blocks $A,B$ and $C$ are $a_A,a_B$ and $a_C$ respectively. Consider string and pulley to be mass-less and friction absent everywhere in the arrangement. Then

1. $2a_A+2a_B+a_C=0$
2. $a_A+a_B+2a_C=0$
3. $a_A+a_B+a_C=0$
4. $2a_A+a_B+a_C=0$

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. 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. 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 figure, the block moves downwards with velocity $u_1$, the wedge moves rightwards with velocity $u_2$. The correct relation between $u_1$ and $u_2$ is

1. $u_2=u_1$
2. $u_2=u_1\sin \alpha$
3. $2u_2\sin \alpha =u_1$
4. $u_2(1+\sin \alpha )=u_1$

Q. Find the constraint relation between acceleration of block 1, 2 and 3.

1. $2a_1+a_2+a_3=0$
2. $a_1+2a_2+a_3=0$
3. $a_1+a_2+2a_3=0$
4. $-2a_1+a_2+a_3=0$

Q. Let θ denote the angular displacement of a simple pendulum oscillating in a vertical plane. If the mass of the bob is m, the tension is the string is mg cos θ
(a) always
(b) never
(c) at the extreme positions
(d) at the mean position.

Q. Figure shows a hemisphere and a supported rod. Hemisphere is moving right with a uniform velocity $v_2$ and the end of rod which is in contact with ground is moving left with a velocity $v_1$. The rate at which the angle $\theta$ is decreasing will be

1. $\frac{(v_1+v_2){\sin }^2\theta }{R\cos \theta }$
2. $\frac{(v_1+v_2)\tan \theta }{R\cos \theta }$
3. $\frac{(v_1+v_2){\cos }^2\theta }{R\sin \theta }$
4. $\frac{(v_1+v_2)\cot \theta }{R\sin \theta }$

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. A simple pendulum is suspended from the ceiling of a car taking a turn of radius 10 m at a speed of 36 km/h. Find the angle made by he string of the pendulum with the vertical if this angle does not change during the turn.

Q. The acceleration of block A is $a$ upwards and the acceleration of block C is $f$ downwards. Find the acceleration of block B. Assume that the string is inextensible and all the pulleys are smooth and light.

1. $\frac{1}{2}(f-a)$, Upwards
2. $\frac{1}{2}(a+f)$, Downwards
3. $\frac{1}{2}(a+f)$, Upwards
4. $\frac{1}{2}(f-a)$, Downwards

Q. A uniform rod of mass m and length l rotates in a horizontal plane with an angular velocity omega about a vertical axis passing through one end. The tension in the rod at a distance x from the axis is

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. If pulleys shown in the diagram are smooth and massless and $a_1$ & $a_2$ are acceleration of blocks of mass $4\text{kg}$ and $8\text{kg}$ respectively. Then find $a_2$.

1. $g/3$
2. $2g/3$
3. $4g/3$
4. $2g$

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. In a given figure, all surfaces are smooth and take $g=10m/s^2$.

1. Acceleration of $2kg$ is $\left(\frac{6g}{17}\right)m/s^2$
2. Acceleration of $4kg$ is $\left(\frac{3g}{17}\right)m/s^2$
3. Acceleration of $6kg$ is $\left(\frac{9g}{17}\right)m/s^2$
4. Tension in string is $\left(\frac{24}{17}g\right)N$

Q. In the figure shown below, the block on the frictionless surface is having acceleration $a$ towards left and the other block is having acceleration $a^{\prime }$. Find the relation between the accelerations of the two blocks.

1. $a=a^{\prime }$
2. $a=2a^{\prime }$
3. $a=\frac{a^{\prime }}{2}$
4. $a=\frac{3}{2}{a}^{\prime }$

Q. Find the velocity $\left(V_A\right)$ of block $A$ in the shown figure.

1. $14\text{m/s}$ downwards.
2. $16\text{m/s}$ upwards.
3. $12\text{m/s}$ downwards.
4. $10\text{m/s}$ upwards.

Q. At a given instant, block A is moving upwards with velocity of $5m/s$, what is velocity of block B at that instant? Assume that the strings are inextensible and pulleys are frictionless.

1. $15m/s\downarrow$
2. $15m/s\uparrow$
3. $10m/s\downarrow$
4. $5m/s\uparrow$

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. In the arrangement shown in the figure, accelerations of blocks $A,B$ and $C$ are $a_A,a_B$ and $a_C$ respectively. Consider string and pulley to be mass-less and friction absent everywhere in the arrangement. Then

1. $2a_A+2a_B+a_C=0$
2. $a_A+a_B+2a_C=0$
3. $a_A+a_B+a_C=0$
4. $2a_A+a_B+a_C=0$