Work Done When Force Is Varying

Q. A body of mass 6kg is under a force which causes displacement in it given by $S=\frac{t^2}{4}$ metres where t is time. The work done by the force in 2 seconds is

1. 6 J
2. 12 J
3. 3 J
4. 9 J

Q. A locomotive of mass m starts moving so that its velocity varies according to the law v = a$\sqrt{s}$, where a is a constant, and s is the distance covered. Find the total work performed by all the forces which are acting on the locomotive during the first t seconds after the beginning of motion.

1. ma⁴t²/8
2. ma³t²/4
3. ma⁴t²/4
4. ma³t²/8

Q.

A particle is moving in a circular path of radius $a$ under the action of an attractive potential $U=-\frac{k}{2r^2}$. Its total energy is

1. Zero

2. $\frac{-3k}{2a^2}$

3. $\frac{-k}{4a^2}$

4. $\frac{k}{2a^2}$

Q.
A small block starts slipping down from a point $B$ on an inclined plane $AB$, which is making an angle $\theta$ with the horizontal, section $BC$ is smooth, and the remaining section $CA$ is rough with a coefficient of friction $\mu$. It is found that the block comes to rest as it reaches the bottom $\left(\text{point}A\right)$ of the inclined plane. If $BC=2AC$, the coefficient of friction is given by $\mu =k\tan \theta$. The value of $k$ is

Q.

A $50kg$ man with a $20kg$load on his head climbs up$20$ steps of $0.25m$ height each. The work done in climbing is

1. $5J$

2. $350J$

3. $100J$

4. $3430J$

Q.

A particle moves from position vector r1=3i^+2j^-6k^ to position vector r2=14i^+13j^+9k^ under the action of force 4i^+j^ +3k^N.Find the work done.

Q. Two persons do the same amount of work. The first person does it in $10\text{s}$ and the second person, in $20\text{s}$. Find the ratio of the power used by the first person to that by the second person.

1. $1:1$
2. $2:1$
3. $1:2$
4. $1:3$

Q. Two pith balls carrying identical charges are suspended from a common point by strings of equal lengths, the equilibrium separation between them is $r$. Now the strings are rigidly clamped at half the height. The equilibrium separation between the balls now will be :

1. $\frac{2r}{\sqrt{3}}$
2. $\frac{2r}{3}$
3. ${\left(\frac{1}{\sqrt{2}}\right)}^2$
4. $\frac{r}{2^{1/3}}$

Q. A force $F$ is acting on an object varies with distance $x$ is shown in the figure given below. The force is in $\text{N}$ and $x$ is in $\text{m}$. The work done by the force in moving the object from $x=0\text{to}x=6\text{m}$ is

1. $4.5\text{J}$
2. $13.5\text{J}$
3. $9.0\text{J}$
4. $18\text{J}$

Q. A position dependent force, $F=(7-2x+3x^2)\text{N}$ acts on a small body of mass $2\text{kg}$ and displaces it from $x=0$ to $x=5\text{m}$. The work done in joules is

1. $270$
2. $135$
3. $35$
4. $70$

Q. A child applies a force $F$ parallel to the $x$-axis to a block moving on a horizontal surface. The $x$-component of the force varies with the $x$-coordinate of the block as shown in figure. The work done by the force $F$ when the block moves from $x=0$ to $x=7\text{m}$ and $x=3\text{m}$ to $4\text{m}$ are (in joules)

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

Q.

Given $\overrightarrow{F}=\left(x{y}^2\right)\hat{i}+\left(x^{2}y\right)\hat{j}N$. The work done by $\overrightarrow{F}$ when a particle is taken along the semicircular path OAB where the coordinates of B are (4, 0) is

1. 

2. 

3. 

4. 

Q.

A particle moves under the effect of a force F = Cx from x = 0 to $x=x_1$ . The work done in the process is

1. 

2. 

3. 

4. Zero

Q.

A uniform chain of length L and mass M overhangs a horizontal table with its two third part on the table. The friction coefficient between the table and the chain is μ. Find the work done by the friction during the period the chain slips off the table.

1. 

2. 

3. 

4. 

Q. A force $F$ acting on a particle varies with the position $x$ as shown in figure. The work done by this force in displacing the particle from $x=-2m$ to $x=0$.

1. $+20\text{J}$
2. $-20\text{J}$
3. $-10\text{J}$
4. $+10\text{J}$

Q.

The distance $x$ moved by a body of mass 0.5 kg due to a force varies with time $t$ as
$x=3t^2+4t+5$
where $x$ is expressed in metre and t in second. What is the work done by the force in the first 2 seconds?

1. 25 J

2. 50 J

3. 100 J

4. 75 J

Q.

Dimensions of kinetic energy are the same as that of

1. Acceleration

2. Force

3. Work

4. Pressure

Q. A force $\overrightarrow{F}=(2x\hat{i}+3y^2\hat{j})\text{N}$ acts on a body and displaces it from the origin $(0,0)$ to $(2\text{m},4\text{m})$.The work done (in $\text{J}$) by the force is

Q. A carpenter is applying a force $\overrightarrow{F}=-1.5x{y}^2\hat{j}$ on a hammer, whose magnitude depends on the position of the tool. The displacement of the tool from the origin to the point $(2,2)\text{m}$. Find the workdone on the hammer, if the displacement is along the straight line $y=x$.

1. $6\text{J}$
2. $0\text{J}$
3. $12\text{J}$
4. $16\text{J}$

Q. A moving coil galvanometer has a coil of area $A$ and number of turns $N$. A magnetic field $B$ is applied on it. Moment of inertia of the coil is $I$ about the axis of rotation.
$\left(a\right)$ Find the torsion constant, $C$ of the spring, if a current $i_0$ produces a deflection of $\frac{\pi }{2}$ in the coil.
$\left(b\right)$ If a charge $Q$ is passed instantaneously through the coil, find the maximum angular deflection, ${\theta }_0$ in it.
$(\text{Maximum torsional energy}=\frac{1}{2}C{\theta }_0^2)$

1. $C=\frac{N{i}_{0}AB}{\pi }$
2. $C=\frac{2N{i}_{0}AB}{\pi }$
3. ${\theta }_0=Q\sqrt{\frac{\pi NAB}{2i_{0}I}}$
4. ${\theta }_0=Q\sqrt{\frac{\pi NAB}{i_{0}I}}$

Q. Three forces starts acting simultaneously on a particle moving with velocity $\overrightarrow{v}$. These forces are represented in magnitude and direction by the three sides of a triangle ABC (as shown). The particle will now move with velocity

1. remaining unchanged
2. Less than
3. Greater than
4. in the direction of the largest force BC

Q.

The lower surface of a cube is fixed. On its upper surface, force is applied at an angle of 30∘ from the surface. The change will be in its:

1. size

2. volume

3. shape

4. both shape and size

Q. A $5kg$ block moves in a striaght line on a horizontal frictionless surface under the influence of a force that varies with position as shown in figure. The workdone by this force as the block moves from the origin to $x=8m$.

1. $35\text{J}$
2. $30\text{J}$
3. $25\text{J}$
4. $20\text{J}$

Q. A force f(bar)=5i(cap)–3j(cap)+2k(cap) moves a particle from r1(bar)=2i+7j+4k to r2=5i+2j+8k. Calculate the work done

Q. A force $\overrightarrow{F}=-k(y\hat{i}+x\hat{j})$ (where $k$ is a positive constant) acts on a particle moving in the $x-y$ plane. Starting from the origin, the particle is taken along the positive $x-$axis to the point $(a,0)$ and then parallel to the $y-$axis to point $(a,a)$. Then, total work done by the force is

1. $-2k{a}^2$
2. $2k{a}^2$
3. $-k{a}^2$
4. $k{a}^2$

Q.

A quantity of air is taken from state a to state b along a path that is a straight line in the P-V diagram. In the previous question If Va = 0.0700 $m^3$, Vb = 0.1100 $m^3$, Pa = 1.00 $\times$ ${10}^5$ Pa, and Pb = 1.40 $\times$${10}^5$ Pa, what is the work W done by the gas in this process? Assume that the gas may be treated as ideal.

1. 4800 J

2. 12000 J

3. 1200 J

4. 5400 J

Q. Coolie $\text{A}$ takes 1 $\text{minute}$ to raise a box through a height of $2\text{m}$. Another coolie $\text{B}$ takes $30\text{sec}$ for the same job and does the same amount of work. Choose the correct option.

1. Coolie $\text{A}$ exerts more power than coolie $\text{B}$
   
2. Coolie $\text{B}$ exerts more power than coolie $\text{A}$
3. Coolie $\text{A}$ exerts same power as coolie $\text{B}$
   
4. None of the above
   

Q. Force acting on a particle varies with displacement as shown in figure. The workdone by this force on the particle from $x=-8mtox=+8m$

1. $-60\text{J}$
2. $100\text{J}$
3. $60\text{J}$
4. $-100\text{J}$

Q. A force F = -K ( yi + aj) (where K is a positive constant) acts on a particle moving in the XY-plane. Starting from the origin, the particle is taken along the positive X-axis to the point (a, 0) and then parallel to the Y-axis to the point (a, a). The total work done by the force F on the particle is

Q.

When brakes are applied to a moving vehicle, the work done by the braking force is

1. Zero

2. Positive

3. Negative

4. Unity