You lift a heavy book from the floor of the room and keep it in the bookshelf having a height of $2 \, m$. In this process,you take $5 \, s$. The work done by you will depend upon:

$A$ force $\vec{F} = 6\hat{i} + 2\hat{j} - 3\hat{k}$ acts on a particle,causing a displacement $\vec{S} = 2\hat{i} - 3\hat{j} + x\hat{k}$. If the work done during this displacement is zero,find the value of $x$.

$A$ particle is moved from $(0, 0)$ to $(a, a)$ under a force $\vec{F} = (3\hat{i} + 4\hat{j})$ through two paths. Path $1$ is $OP$ and path $2$ is $OQP$. Let $W_1$ and $W_2$ be the work done by this force in these two paths respectively. Then

$A$ body of mass $2 \ kg$ is moving along the $x$-direction such that its displacement as a function of time is given by $x(t) = \alpha t^2 + \beta t + \gamma \ m$, where $\alpha = 1 \ m/s^2$, $\beta = 1 \ m/s$, and $\gamma = 1 \ m$. The work done on the body during the time interval $t = 2 \ s$ to $t = 3 \ s$ is . . . . . . $J$.

$A$ body constrained to move along the $z$-axis of a coordinate system is subject to a constant force $F$ given by $F = -\hat{i} + 2\hat{j} + 3\hat{k} \text{ N}$,where $\hat{i}, \hat{j}, \hat{k}$ are unit vectors along the $x, y,$ and $z$-axes of the system respectively. What is the work done (in $J$) by this force in moving the body a distance of $4 \text{ m}$ along the $z$-axis?

$A$ sphere of mass $m$ is tied to one end of a string of length $l$ and rotated through the other end along a horizontal circular path with speed $v$. The work done in a full horizontal circle is