Work Done, Varying Force and Path

Q. A bullet is fired from a gun. The force on the bullet is given by F = 600 - 2 $\times$ ${10}^5$ t, where F is in Newton and t in seconds. The force on the bullet becomes zero as soon as it leaves the barrel. What is the average impulse imparted to the bullet

1. 9 Ns
2. Zero
3. 0.9 Ns
4. 1.8 Ns

Q.

A body of mass 3 kg is under a force, which causes a displacement in it is given by $S=\frac{t^3}{3}$ (in m). Find the work done by the force in first 2 seconds

1. 2 J
2. 3.8 J
3. 5.2 J
4. 24 J

Q. A block of $10\text{kg}$ is pulled in a vertical plane along a frictionless surface in the form of an arc of a circle of radius $10\text{m}$. A force of $200\text{N}$ is applied such that tension in the string always remains along the tangential direction as shown in figure. If the block started from rest at $A$, the velocity at $B$ would be:
Consider $\frac{\pi }{3}=1$ for calculation and take $g=10{\text{m/s}}^2$)

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

Q. Velocity - time graph of a particle of mass $2\text{kg}$ moving in a straight line is as shown in figure. Work done by all the forces on the particle is

1. $400\text{J}$
2. $-400\text{J}$
3. $-200\text{J}$
4. $200\text{J}$

Q. The work done in joules in increasing the extension of a spring of stiffness 10 N/cm from 4 cm to 6 cm is:

1. 1
2. 10
3. 50
4. 100

Q. The force exerted by a compression device is given by $F\left(x\right)=kx(x-l)$, where $l$ is the maximum possible compression, $x$ is the compression and $k$ is a constant. The work required to compress the device by a distance $d$ will be maximum when

1. $d=\frac{l}{4}$
2. $d=\frac{l}{2}$
3. $d=\frac{l}{\sqrt{2}}$
4. $d=l$

Q. A machine in an amusement park, consists of a cage at the end of one arm, hinged at $O$. The cage revolves along a vertical circle $\text{(ABCDEFGH)}$ of radius $r$ about its hinge $O$, at constant linear speed $v=\sqrt{gr}$. The cage is so attached that the man of weight $W$ standing on a weighing machine, inside the cage, is always vertical. Then which of the following is/are correct.

1. the weight reading at $\text{A}$ is greater than the weight reading at $\text{E}$ by $2W$.
2. the weight reading at $\text{G}$ is $W$.
3. the ratio of the weight reading at $\text{E}$ to that at $\text{A}$ is $0$.
4. the ratio of the weight reading at $\text{A}$ to that at $\text{C}$ is $2$.

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 the point $(a,a)$. Find the total work done by the forces $\overrightarrow{F}$ on the particle.

1. Ka²

2. -2 Ka²

3. 2 ka²

4. 

Q. A spring which is initially in its unstretched condition, is first stretched by a length x and then again by a further length x. The work done in the first case is $W_1$ and in the second case is $W_2$. Then:

1. $W_2=W_1$
2. $W_2=2W_1$
3. $W_2=3W_1$
4. $W_2=4W_1$

Q. A Force $\overrightarrow{F}=-2k[y\hat{i}+x\hat{j}]N$ , where $k$ is 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 $y-$ axis to the point $(a,a)$. The total work done by the force $\overrightarrow{F}$ on the particle is

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

Q. A spring of spring constant $5\times {10}^3\text{N/m}$ is stretched initially by $5cm$ from the unstretched position. The work required to further stretch the spring by another $5cm$ is

1. $6.25\text{J}$
2. $12.50\text{J}$
3. $18.75\text{J}$
4. $25.00\text{J}$

Q. Two wires $\text{AC}$ and $\text{BC}$ are tied at $\text{C}$ to a small sphere of mass $5\text{kg}$, which revolves a constant speed $v$ in the horizontal circle of radius $1.6\text{m}$. If both wires are to remain taut and if the tension in either of the wires is not to exceed $60\text{N}$, then minimum value of $v$ is
[Take, $g=10{\text{m/s}}^2$]

1. $3.0\text{m/s}$
2. $4.0\text{m/s}$
3. $8.2\text{m/s}$
4. $3.96\text{m/s}$

Q.

A person is holding a bucket by applying a force of $10N$. He moves a horizontal distance of $5m$ and then climbs up a vertical distance of $10m$. The total work done by him is equal to

1. $50J$
2. $150J$
3. $100J$
4. $200J$

Q. A force $F=-K(y\hat{i}-x\hat{j})$, (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 plane $(a,0)$ and then parallel to the $Y-$axis to the point $(a,a)$. The total work done by the force $F$ on the particle is

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

Q. A block of mass $10$ kg, moving in x direction with a constant speed of $10m{s}^{-1}$, is subjected to a retarding force $F=0.1xJ/m$ during its travel from $x=-20m$ to $30m$. Its final KE will be

1. 275 J
2. 250 J
3. 475 J
4. 450 J

Q. The force exerted by a compression device is given by $F\left(x\right)=kx(x-l)$, where $l$ is the maximum possible compression, $x$ is the compression and $k$ is a constant. The work required to compress the device by a distance $d$ will be maximum when

1. $d=\frac{l}{4}$
2. $d=\frac{l}{\sqrt{2}}$
3. $d=\frac{l}{2}$
4. $d=l$

Q. Assertion :A person working on a horizontal road with a load on his head does no work. Reason: No work is said to be done, if directions of force and displacement of load are perpendicular to each other.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. A bullet is fired from a gun. The force on the bullet is given by $F=600-2\times {10}^{5}t$, where F is in newtons and t in seconds. The force on the bullet becomes zero as soon as it leaves the barrel. What is the average impulse imparted to the bullet

1. 0.9 Ns
2. 1.8 Ns
3. 9 Ns
4. Zero

Q. If a light rod is rotated by angle $\theta$ and torque equation is given by $\iota =k\theta$. If rod is rotated by angle ${\theta }_0$ then the maximum tension in the rod is

1. $\frac{3k({\theta }_0{)}^2}{\ell }$
2. $\frac{2k({\theta }_0{)}^2}{\ell }$
3. $\frac{k({\theta }_0{)}^2}{\ell }$
4. $\frac{k({\theta }_0{)}^2}{2\ell }$

Q. The potential energy of a spring when stretched through a distance $x$ is $10J$. What is the amount of work done on the same spring to stretch it through an additional distance $x$?

1. $10J$
2. $20J$
3. $40J$
4. $30J$

Q. A bullet of mass ${10}^{-3}\text{kg}$ strikes an obstacle and moves at ${60}^{\circ }$ to its original direction. If its speed also changes from $20\text{m/s}$ to $10\text{m/s}$. Find the magnitude of impulse acting on the bullet.

1. $\sqrt{2}\times {10}^{-2}\text{Ns}$
   
2. $\sqrt{3}\times {10}^{-2}\text{Ns}$
   
3. $\sqrt{5}\times {10}^{-2}\text{Ns}$
   
4. $2\times {10}^{-2}\text{Ns}$

Q. A force $F=Ai+Bj$ is applied to a point whose radius vector relative to the origin of coordinates $O$ is equal to $r=ai+bj$, where $a,b,A,B$ are constants, and $i,j$ are the unit vector of the $x$ and $y$ axes $a,b,A,B$ are constants, Find the arm $l$ of the force $F$ relative to the point $O$.

Q. Consider a force $\overrightarrow{F}=-x\hat{i}+y\hat{j}$. The work done by this force in moving a particle from point $A(1,0)$ to $B(0,1)$ along the line segment is?
(all quantities are in SI units)

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

Q. An objects of mass $40kg$ having velocity $4\hat{i}m/s$ collides with another object of mass $60kg$ having velocity $2\hat{i}m/s$. If collision is pefectly inelastic, then the loss of mechanical energy is

1. $24J$
2. $96J$
3. $48J$
4. $6J$

Q. A spring of force constant $10$ N/m has an initial stretch $0.20$ m. In changing the stretch to $0.25$ m, the increase in potential energy is about

1. $0.5$ joule
2. $0.1$ joule
3. $0.2$ joule
4. $0.3$ joule

Q. A bullet is fired from a gun. The force on the bullet is given by F = 600 - 2 $\times$ ${10}^5$ t, where F is in Newton and t in seconds. The force on the bullet becomes zero as soon as it leaves the barrel. What is the average impulse imparted to the bullet

1. 9 Ns
2. Zero
3. 0.9 Ns
4. 1.8 Ns

Q.

A force of 50 dynes is acted on a body of mass 5 g which is at rest for an interval of 3 seconds, then impulse is

1. $0.15\times {10}^{-3}N-s$

2. $0.98\times {10}^{-3}N-s$

3. $1.5\times {10}^{-3}N-s$

4. $2.5\times {10}^{-3}N-s$

Q. A spring of spring constant $5\times {10}^3\text{N/m}$ is stretched initially by $5cm$ from the unstretched position. The work required to further stretch the spring by another $5cm$ is

1. $6.25\text{J}$
2. $12.50\text{J}$
3. $18.75\text{J}$
4. $25.00\text{J}$

Q. A force $\overrightarrow{F_1}=A\overrightarrow{j}$ is applied to a point whose radius vector is $\overrightarrow{r_1}=a\overrightarrow{i}$, while a force $\overrightarrow{F_2}=B\overrightarrow{i}$ is applied to the point whose radius vector is $\overrightarrow{r_2}=b\overrightarrow{j}$. Both radius vectors are determined relative to the origin of coordinates $O$. It is given $\overrightarrow{i}$ and $\overrightarrow{j}$ are the unit vectors of the x and y axes and $a$, $b$, $A$, $B$ are constants. Find the arm $l$ of the resultant force relative to the point $O$.

1. $l=\frac{aA-bB}{\sqrt{A^2+B^2}}$
2. $l=\frac{ab+AB}{\sqrt{A^2+B^2}}$
3. $l=\frac{ab-AB}{\sqrt{A^2-B^2}}$
4. None of these

Q. A bullet is fired from a gun, the force on the bullet is given by $F=600-2\times {10}^{5}t$ where $F$ is in newton and $t$ in second. The force on the bullet becomes zero as soon as it leaves the barrel. What is the average impulse imparted to the bullet?

1. $0$
2. $1.8N-s$
3. $9N-s$
4. $0.9N-s$