Chapter 10.7, Problem 52PS

Solution

(a)

To find: Simplify the equation to show $g=978+5.17{\sin }^{2}L-0.007\sin 2L$

Explanation is given.

Given information:

The given equation is $g=978+5.17{\sin }^{2}L-0.014\sin L\cos L$ .

In order to change this equation apply the double angle formula $\sin 2\theta =2\sin \theta \cos \theta$

The given equation can be rewritten as:

  $g=978+5.17{\sin }^{2}L-(0.007\cdot 2)\sin L\cos L\mathrm{.....}(1)$

Now $2\sin L\cos L=\sin 2L$

Substitute this value in the equation $\left(1\right)$ it becomes:

  $g=978+5.17{\sin }^{2}L-0.007\sin 2L$

This is the required form.

(b)

To find: Graph the function from part (b).

Given information:

The given equation is :

  $g=978+5.17{\sin }^{2}L-0.007\sin 2L$

In order to graph this equation substitute different value of $L$ and find corresponding value of $g$ .

Let $L=\frac{\pi }{2}$ substitute this value in $g=978+5.17{\sin }^{2}L-0.007\sin 2L$ gives:

  $\begin{array}{l}g=978+5.17{\sin }^2\left(\frac{\pi }{2}\right)-0.007\sin 2\frac{\pi }{2}\\ g=978+5.17{\left(1\right)}^2-0.007\left(0\right)\sin \left(\frac{\pi }{2}\right)=1,\sin \pi =0\\ g=978+5.17\\ g=983.17\end{array}$

Let $L=\frac{\pi }{4}$ substitute this value in $g=978+5.17{\sin }^{2}L-0.007\sin 2L$ gives:

  $\begin{array}{l}g=978+5.17{\sin }^2\left(\frac{\pi }{4}\right)-0.007\sin 2\frac{\pi }{4}\\ g=978+5.17{\left(\frac{1}{\sqrt{2}}\right)}^2-0.007\left(\frac{\pi }{2}\right)\sin \left(\frac{\pi }{2}\right)=1,\sin \left(\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}\\ g=978+5.17\frac{1}{2}-0.007\\ g=980.528\end{array}$

Similarly, more value can be found and then draw them on graph will gives the required graph.

The graph is given below:

  

(c)

To find: From the graph find the acceleration due to gravity when latitude is ${45}^0,{30}^0,0^0$

Given information:

The graph found in part (b) is below:

  

From the graph when latitude is $0^0$ the acceleration is $978$ ,When latitude is ${45}^0$ the acceleration is $980.528$ ,When latitude is ${30}^0$ the acceleration is $979.28$