Work by a Variable Force

A particle of mass $M$ moves along a horizontal $x$ axis from $x = 0$ to $x = L$. The coefficient of kinetic friction varies as a function of $x$ as $\mu_k(x) = \mu_0 - \alpha x$, where $\mu_0, \alpha$ are constants of appropriate dimensions, so that $\mu_k(L) = 0$. The total work done by the frictional force during the motion is $n\mu_0 MgL$, where $g$ is the acceleration due to gravity. The value of $n$ is:

A force $F = \alpha + \beta x^2$ acts on an object in the x -direction. The work done by the force is 5 J when the object is displaced by 1 m. If the constant $\alpha = 1\text{ N}$ then $\beta$ will be

A force $f = x^2y\hat{i} + y^2\hat{j}$ acts on a particle in a plane $x + y = 10$. The work done by this force during a displacement from $(0, 0)$ to $(4\text{ m}, 2\text{ m})$ is _________ Joule (round off to the nearest integer)

An object moving along horizontal $x$-direction with kinetic energy $10\text{ J}$ is displaced through $x = (3\hat{i})\text{ m}$ by the force $\vec{F} = (-2\hat{i} + 3\hat{j})\text{ N}$. The kinetic energy of the object at the end of the displacement $x$ is

A force $\vec{F} = (2 + 3x)\hat{i}$ acts on a particle in the $x$ direction where F is in newton and $x$ is in meter. The work done by this force during a displacement from $x = 0$ to $x = 4\text{ m}$, is __________ J.

A force $F = \left(5 + 3y^2\right)$ acts on a particle in the $y$-direction, where $F$ is in newton and $y$ is in meter. The work done by the force during a displacement from $y = 2\text{ m}$ to $y = 5\text{ m}$ is ___________ J.

A block of mass $10\text{ kg}$ is moving along $x$-axis under the action of force $F = 5x\text{ N}$. The work done by the force in moving the block from $x = 2\text{ m}$ to $4\text{ m}$ will be __________ J.

A particle of mass $10\text{ g}$ moves in a straight line with retardation $2x$, where $x$ is the displacement in SI units. Its loss of kinetic energy for above displacement is $\left(\frac{10}{x}\right)^{-n}$ J. The value of $n$ will be __________

An object of mass 'm' initially at rest on a smooth horizontal plane starts moving under the action of force F = 2N. In the process of its linear motion, the angle $\theta$ (as shown in figure) between the direction of force and horizontal varies as $\theta = kx$, where k is a constant and $x$ is the distance covered by the object from its initial position. The expression of kinetic energy of the object will be $E = \frac{n}{k}\sin\theta$. The value of n is ___________.

Arrange the four graphs in descending order of total work done; where W1, W2, W3 and W4 are the work done corresponding to figure a, b, c and d respectively.

A block of mass 2 kg moving on a horizontal surface with speed of 4 ms${}^{-1}$ enters a rough surface ranging from x = 0.5 m to x = 1.5 m. The retarding force in this range of rough surface is related to distance by F = $-$ kx where k = 12 Nm${}^{-1}$. The speed of the block as it just crosses the rough surface will be:

A particle experiences a variable force $\vec{F} = \left(4x\hat{i} + 3y^2\hat{j}\right)$ in a horizontal x-y plane. Assume distance in meters and force is newton. If the particle moves from point (1, 2) to point (2, 3) in the x-y plane, then Kinetic Energy changes by:

A force of F = (5y + 20)$\hat{j}$ N acts on a particle. The work done by this force when the particle is moved from y = 0 m to y = 10 m is ___________ J.

A person pushes a box on a rough horizontal platform surface. He applies a force of 200 N over a distance of 15 m. Thereafter, he gets progressively tired and his applied force reduces linearly with distance to 100 N. The total distance through which the box has been moved is 30 m. What is the work done by the person during the total movement of the box?

Consider a force $\vec{F} = -x\hat{i} + y\hat{j}$. 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)

A force F = 20 + 10y acts on a particle in y-direction where F is in newton and y in meter. Work done by this force to move the particle from y = 0 to y = 1 m is:

An object of mass 500 g, initially at rest, is acted upon by a variable force whose X-component varies with X in the manner shown. The velocities of the object at the points X = 8 m and X = 12 m, would have the respective values of (nearly)

A particle moves in one dimension from rest under the influence of a force that varies with the distance travelled by the particle as shown in the figure. The kinetic energy of the particle after it has travelled 3m is:

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

When a rubber-band is stretched by a distance $x$, it exerts restoring force of magnitude $F = ax + bx^2$ where $a$ and $b$ are constants. The work done in stretching the unstretched rubber-band by $L$ is:

Force F on a particle moving in a straight line varies with distance d as shown in figure.

The work done on the particle during its displacement of 12 m is

A force F acting

on an object varies with distance x as shown here. The force is in N and x in m. The work done by the force in moving the object from x = 0 to x = 6 m is

A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement $x$ is proportional to

A body, constrained to move in $y$-direction, is subjected to a force given by $\vec{F} = (-2\hat{i} + 15\hat{j} + 6\hat{k})\text{ N}$. The work done by this force in moving the body through a distance of $10\hat{j}\text{ m}$ along $y$-axis, is

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 joule is

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 joule is