Work Energy and...

A simple pendulum is released from rest with the string in horizontal position. The vertical component of the velocity of the bob becomes maximum, when the string makes an angle $$\displaystyle \theta $$ with the vertical. The angle $$\displaystyle \theta $$ is equal to

A force $$\displaystyle F= -\frac{k}{x^{2}}\left ( x\neq 0 \right )$$ acts on a particle in x-direction. Find the work done by this force in displacing the particle from. $$\displaystyle x= +a\:$$to$$\:x= +2a.$$ Here, $$k$$ is a positive constant.

An object is displaced from position vector $$\displaystyle \vec{r}_{1}= \left ( 2\hat{i}+3\hat{j} \right )m\:$$ to$$ \:\vec{r}_{2}= \left ( 4\hat{i}+6\hat{j} \right )$$ m under a force $$\displaystyle \vec{F}= \left ( 3x^{2}\hat{i}+2y\hat{j} \right )N.$$ Find the work done by this force.

A particle is moving in a circular path in the vertical plane. It is attached at one end of a string of length $$l$$ whose other end is fixed. The velocity at lowest point is $$u$$. The tension in the string is $$\displaystyle \vec{T}$$ and acceleration of the particle is $$\displaystyle \vec{a}$$ at any position. Then $$\displaystyle \vec{T}.\vec{a}$$ is zero at highest point if

A cutting tool under microprocessor control has several forces acting on it. One force is $$\vec{F}=-\alpha xy^2\hat{j}$$, a force in the negative y-direction whose magnitude depend on the position of the tool. The constant is $$\alpha  = 2.50\ N/m^3$$. Consider the displacement of the tool from the origin to the point  $$x = 3.00 \,m, y = 3.00 \,m$$. Calculate the work done on the tool by $$\vec{F}$$ if the tool is first moved out along the x-axis to the point $$x = 3.00m, \:y= 0m $$ and then moved parallel to the y-axis to $$x = 3.00m, y = 3.00 \,m$$.

A particle of mass $$m$$ is suspended by a string of length $$l$$ from a fixed rigid support. A sufficient horizontal velocity $$\displaystyle v_{0}= \sqrt{3gl}$$ is imparted to it suddenly. Calculate the angle made by the string with the vertical when the acceleration of the particle is inclined to the string by $$\displaystyle 45^{\circ}.$$

The sphere at A is given a downward velocity $$\displaystyle v_{0}$$ of magnitude $$5 m/s$$ and swings in a vertical plane at the end of a rope of length $$l=2 m$$ attached to a support at $$O$$. Determine the angle $$\displaystyle \theta $$ at which the rope will break, knowing that it can withstand a maximum tension equal to twice the weight of the sphere.

just after it comes in contact with the peg.

just before the sphere comes in contact with the peg.

if AB is a massless rod,