Chapter 8, Problem 100QAP

To determine

Angular speed of the dough when the chef catch the pizza.

Answer to Problem 100QAP

  ${\omega }_{after}=4.298rad/s$

Explanation of Solution

Given info:

Weight of pizza crust $=$

  $0.500kg$

Initial angular speed of dough before expanding $={\omega }_{before}=5.20rad/s$

Diameter of pizza before expanding $=d_{before}=20cm$

So, $r_{before}=10cm$

Diameter of pizza after expanding $=d_{after}=22cm$

So, $r_{after}=11cm$

Formula used:

Angular momentum,

  $L_z=I{\omega }_z$ ( $L_z=$ Angular momentum, $I=$ moment of inertia, ${\omega }_z=$ angular velocity)

Conservation of angular momentum,

  $L_{before}=L_{after}$

  $I_{before}{\omega }_{before}=I_{after}{\omega }_{after}$

Calculation:

Moment of inertia of the dough before the expanding,

  $I_{before}=\frac{1}{2}m{\left(r_{before}\right)}^2$

Angular momentum of the dough before the expanding,

  $L_{before}=I_{before}{\omega }_{before}$

  $L_{before}=\frac{1}{2}m{\left(r_{before}\right)}^2{\omega }_{before}$$\mathrm{...}(1)$

Moment of inertia of the dough after the expanding,

  $I_{after}=\frac{1}{2}m{\left(r_{after}\right)}^2$

Angular momentum of the dough before the expanding,

  $L_{after}=I_{after}{\omega }_{after}$

  $L_{after}=\frac{1}{2}m{\left(r_{after}\right)}^2{\omega }_{after}$$\mathrm{...}(2)$

According to the Conservation of angular momentum,

  $L_{before}=L_{after}$

  $I_{before}{\omega }_{before}=I_{after}{\omega }_{after}$

Substituting $(1),(2)$ expressions,

  $\frac{1}{2}m{\left(r_{before}\right)}^2{\omega }_{before}=\frac{1}{2}m{\left(r_{after}\right)}^2{\omega }_{after}$

  ${\left(r_{before}\right)}^2{\omega }_{before}={\left(r_{after}\right)}^2{\omega }_{after}$

Substituting the given values, we get

  ${\left(10\times {10}^{-2}m\right)}^2(5.2rad/s)={\left(11\times {10}^{-2}m\right)}^2{\omega }_{after}$

  $\left(100\times {10}^{-4}{m}^2\right)(5.2rad/s)=\left(121\times {10}^{-4}{m}^2\right){\omega }_{after}$

  $100(5.2rad/s)=121{\omega }_{after}$

  ${\omega }_{after}=\frac{100(5.2rad/s)}{121}$

${\omega }_{after}=4.298rad/s$

Conclusion:

Thus, angular speed of the dough, after the expanding is $4.298rad/s$.