91影视

The mass of the object is m = 0.25 kg

The force acting on the object is$\stackrel{鈬赌}{F}=4.0\hat{j}N$

The position vector is$\stackrel{鈬赌}{r}=\left(2.0m\right)\hat{i}-\left(2.0m\right)\hat{k}$

The velocity vector is$\stackrel{鈬赌}{v}=\left(-5.0m/s\right)\hat{i}+\left(5.0m/s\right)\hat{k}$

The problem deals with the calculation of angular momentum. The angular momentum of a rigid object is product of the moment of inertia and the angular velocity. It is analogous to linear momentum. It also deals with the torque. It is a measure of the force that can cause an object to rotate about an axis

Formulae:

$$\begin{gathered} \stackrel{鈬赌}{l}=m\left(\stackrel{鈬赌}{r}脳\stackrel{鈬赌}{v}\right) \\ \stackrel{鈬赌}{蟿}=\stackrel{鈬赌}{r}脳\stackrel{鈬赌}{F} \end{gathered}$$

Let position vector is $\stackrel{鈬赌}{r}=x\hat{i}+y\hat{j}+z\hat{k}$and velocity vector is $\stackrel{鈬赌}{v}=v_x\hat{i}+v_y\hat{j}+v_z\hat{k}$.

The cross product of the position vector and velocity vector is,

$\stackrel{鈬赌}{r}脳\stackrel{鈬赌}{v}=\left(y{v}_z-z{v}_y\right)\hat{i}+\left(z{v}_x-x{v}_z\right)\hat{j}+\left(x{v}_y-y{v}_x\right)\hat{k}$

In the given position and velocity vector, y = 0 and v_y = 0m/s. Then,

$\stackrel{鈬赌}{r}脳\stackrel{鈬赌}{v}=\left(z{v}_x-x{v}_z\right)\hat{j}$

The angular momentum of the object with position vector and velocity vector is,

$\begin{array}{rcl}\stackrel{鈬赌}{l}& =& m\left(\stackrel{鈬赌}{r}脳\stackrel{鈬赌}{v}\right)\\ & =& m\left(z{v}_x-x{v}_z\right)\hat{j}\\ & =& 0.25kg\left[\left(-2.0m\right)\left(-5.0m/s\right)-\left(2.0m\right)\left(5.0m/s\right)\right]\hat{j}\\ & =& 0kg\frac{m^2}{s}\end{array}$

To find torque acting on the object, force acting on it will be $\stackrel{鈬赌}{F}={\stackrel{鈬赌}{F}}_x\hat{i}+{\stackrel{鈬赌}{F}}_y\hat{j}+{\stackrel{鈬赌}{F}}_z\hat{k}$.

The expression of torque is,

$\begin{array}{rcl}\stackrel{鈬赌}{蟿}& =& \stackrel{鈬赌}{r}脳\stackrel{鈬赌}{F}\\ & =& \left(y{F}_z-z{F}_y\right)\hat{i}+\left(z{F}_x-x{F}_z\right)\hat{j}+\left(x{F}_y-y{F}_x\right)\hat{k}\end{array}$

In the given position and force vector, y= 0m and

$\begin{array}{rcl}{F}_x& =& F_z\\ & =& 0N\\ \stackrel{鈬赌}{蟿}& =& \left(-z{F}_y\right)\hat{i}+\left(x{F}_y\right)\hat{k}\\ & =& -\left(-2.0m\right)\left(4.0N\right)\hat{i}+\left(2.0m\right)\left(4.0N\right)\hat{k}\\ & =& \left(8.0N.m\right)\hat{i}+\left(8.0N.m\right)\hat{k}\end{array}$