91影视

The Energy Integral Lemma:

Let y(t) be a solution to the differential equation$\mathrm{y}=\mathrm{f}\left(\mathrm{y}\right)$, where f(y) is a continuous function that does not depend on y鈥� or the independent variable t. Let F(y) is an indefinite integral of$\mathrm{f}\left(\mathrm{y}\right)$ , that is,$\mathrm{f}\left(\mathrm{y}\right)=\frac{\mathrm{d}}{\mathrm{dy}}\mathrm{F}\left(\mathrm{y}\right)$ . Then the quantity $\mathrm{E}\left(\mathrm{t}\right):=\frac{1}{2}\mathrm{y}\text{'}{\left(\mathrm{t}\right)}^2-\mathrm{F}\left(\mathrm{y}\left(\mathrm{t}\right)\right)$ is constant; i.e., $\frac{\mathrm{d}}{\mathrm{dt}}\mathrm{E}\left(\mathrm{t}\right)=0$.

Change of angular momentum:

data-custom-editor="chemistry" $\mathrm{m}{\mathrm{鈩�}}^2\mathrm{胃}=-\mathrm{鈩�}\mathrm{尘驳蝉颈苍胃}.......\left(1\right)$

鈥︹�� (1)

Newton鈥檚 rotational law: The rate of change of angular momentum is equal to torque.

Given that, the figure of the pendulum is shown below.

To prove: angular momentum $=\mathrm{m}{\mathrm{鈩�}}^2\frac{\mathrm{诲胃}}{\mathrm{dt}}$鈥︹�� (2)

From the figure, we can see that the angular velocity is,$\mathrm{v}=\frac{\mathrm{诲胃}}{\mathrm{dt}}$ .

Then, find the angular momentum.

$$\begin{gathered} \mathrm{L}=\mathrm{lmv} \\ =\mathrm{lm}脳\mathrm{l}脳\frac{\mathrm{诲胃}}{\mathrm{dt}} \\ ={\mathrm{ml}}^2\frac{\mathrm{诲胃}}{\mathrm{dt}} \end{gathered}$$

So, angular momentum$=\mathrm{m}{\mathrm{鈩�}}^2\frac{\mathrm{诲胃}}{\mathrm{dt}}$ is true.

To prove: Torque$=-\mathrm{鈩�}\mathrm{尘驳蝉颈苍胃}$.

From the given figure we can see that the restoring force is,$\mathrm{F}=\mathrm{尘驳蝉颈苍胃}$ .

Then, the relation between torque T and force F is,role="math" localid="1663957923230" $\mathrm{T}=\mathrm{lF}$ .

Substitute the value of force in torque.

$\mathrm{T}=\mathrm{l尘驳蝉颈苍胃}$

Since the force and $\mathrm{胃}$ are in opposite directions. So, $\mathrm{T}=-\mathrm{l尘驳蝉颈苍胃}$

Hence proved.

Given that, Newton鈥檚 rotational law.

That is, the rate of change of angular momentum is equal to torque.

The angular momentum is,$\mathrm{L}={\mathrm{ml}}^2\frac{\mathrm{诲胃}}{\mathrm{dt}}$ . Then, differentiate with respect to t.

$\frac{\mathrm{dL}}{\mathrm{dt}}={\mathrm{ml}}^2\frac{{\mathrm{d}}^2\mathrm{胃}}{{\mathrm{dt}}^2}$

Then,

${\mathrm{ml}}^2\frac{{\mathrm{d}}^2\mathrm{胃}}{{\mathrm{dt}}^2}=-\mathrm{l尘驳蝉颈苍胃}$

So, which is similar to equation (1). Then, the statement is true.

Hence proved