College Physics by Openstax Chapter 3 Problem 9: Verification of the Vector Sum

Problem:

Show that the sum of the vectors discussed in Example 3.2 gives the result shown in Figure 3.24.

Solution:

So, we are given the two vectors shown below.

If we use the graphical method of adding vectors, we can join the two vectors using head-tail addition and come up with the following:

***Figure 3.9B: Vectors A and B added graphically***

The resultant is drawn from the tail of the first vectors (the origin) to the head of the last vector. The resultant is shown in red in the figure below.

Solve for the value of the angle 𝛼 by geometry.

\alpha = 66^\circ +\left( 180^\circ-112^\circ \right) = 134^\circ

Solve for the magnitude of the resultant using cosine law.

\begin{align*} R^2 & = A^2+B^2-2AB\cos \alpha \\ R & = \sqrt{A^2+B^2-2AB\cos \alpha} \\ R & = \sqrt{\left( 27.5 \ \text{m} \right)^2+\left( 30.0 \ \text{m} \right)^2-2\left( 27.5\ \text{m} \right)\left( 30.0\ \text{m} \right) \cos 134^\circ} \\ R & =52.9380 \ \text{m} \\ R & = 52.9 \ \text{m} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

Solve for 𝛽 using sine law.

\begin{align*} \frac{\sin \beta}{B} & = \frac{\sin \alpha}{R} \\ \beta & = \sin ^{-1} \left( \frac{B \sin \alpha }{R} \right) \\ \beta & = \sin ^{-1} \left( \frac{30.0\ \text{m} \sin 134^\circ}{52.9380 \ \text{m}} \right) \\ \beta & = 24.0573^\circ \end{align*}

Finally, solve for 𝜃.

\theta = 66^\circ+24.0573^\circ = 90.1^\circ \ \qquad \ {\color{Orange} \left( \text{Answer} \right)}

The result is in conformity with that in figure 3.24 shown on the question shown above.

College Physics Chapter 3 Problems

Problem 1	Problem 2	Problem 3	Problem 4	Problem 5
Problem 6	Problem 7	Problem 8	Problem 9	Problem 10
Problem 11	Problem 12	Problem 13	Problem 14	Problem 15
Problem 16	Problem 17	Problem 18	Problem 19	Problem 20
Problem 21	Problem 22	Problem 23	Problem 24	Problem 25
Problem 26	Problem 27	Problem 28	Problem 29	Problem 30
Problem 31	Problem 32	Problem 33	Problem 34	Problem 35
Problem 36	Problem 37	Problem 38	Problem 39	Problem 40
Problem 41	Problem 42	Problem 43	Problem 44	Problem 45
Problem 46	Problem 47	Problem 48	Problem 49	Problem 50
Problem 51	Problem 52	Problem 53	Problem 54	Problem 55
Problem 56	Problem 57	Problem 58	Problem 59	Problem 60
Problem 61	Problem 62	Problem 63	Problem 64	Problem 65
Problem 66	Problem 67	Problem 68	Problem 69	Problem 70