(a) What is a vector quantity? (b) Three vectors \(3\ \text{ms}^{-1}\) N \(45^\circ\) W, \(12\ \text{ms}^{-1}\) W and \(5\ \text{ms}^{-1}\) S act at a point...
Question 1 Report
(a) What is a vector quantity?
(b) Three vectors \(3\ \text{ms}^{-1}\) N \(45^\circ\) W, \(12\ \text{ms}^{-1}\) W and \(5\ \text{ms}^{-1}\) S act at a point.
(i) Sketch a vector diagram to illustrate the given information.
(ii) Calculate the resultant of the vectors.
(c) In a laboratory experiment to determine the force constant of a spiral spring, the mass or, the spring was varied and the corresponding extensions were measured and recorded as shown in the table below.
| Mass M/g | Weight W/N | Extension e/cm |
| 50 | 6.5 | |
| 100 | 11.0 | |
| 150 | 15.0 | |
| 200 | 20.0 | |
| 250 | 25.0 |
(i) Copy and complete the table. (Take \(g = 10\ \text{ms}^{-2}\))
(ii) Plot a graph with weight, W, on the vertical axis and extension, e, on the horizontal axis.
(iii) Using the graph, determine the force constant of the spring.
(iv) Determine the natural length of the spring if its length was 38.0 cm when loaded with 250 g mass.
Answer Details
(a) A vector quantity is a physical quantity which has both magnitude and direction. Examples include velocity, force and displacement.
(b)(i) Vector diagram
(b)(ii) Resultant of the vectors
Take east as the positive x-direction and north as the positive y-direction.
| Vector | x-component / m s-1 | y-component / m s-1 |
|---|---|---|
| 3 m s-1, N45°W | \(-3\sin45^\circ=-2.12\) | \(+3\cos45^\circ=+2.12\) |
| 12 m s-1, W | \(-12.00\) | \(0\) |
| 5 m s-1, S | \(0\) | \(-5.00\) |
| Resultant components | \(-14.12\) | \(-2.88\) |
\[R=\sqrt{(-14.12)^2+(-2.88)^2}=14.4\ \text{m s}^{-1}\]
\[\theta=\tan^{-1}\left(\frac{2.88}{14.12}\right)=11.5^\circ\]
Hence, the resultant is \(14.4\ \text{m s}^{-1}\), \(11.5^\circ\) south of west, or equivalently S78.5°W.
(c)(i) Completed table
Since \(W=Mg\), with \(g=10\ \text{m s}^{-2}\):
| Mass, \(M\) / g | Weight, \(W\) / N | Extension, \(e\) / cm |
|---|---|---|
| 50 | 0.5 | 6.5 |
| 100 | 1.0 | 11.0 |
| 150 | 1.5 | 15.0 |
| 200 | 2.0 | 20.0 |
| 250 | 2.5 | 25.0 |
(c)(ii) Graph of weight against extension
(c)(iii) The force constant, \(k\), is the gradient of the graph:
Using two widely separated points on the line of best fit, \((10.0\ \text{cm},0.903\ \text{N})\) and \((25.0\ \text{cm},2.530\ \text{N})\),
\[k=\frac{\Delta W}{\Delta e}=\frac{2.530-0.903}{25.0-10.0}=0.1085\ \text{N cm}^{-1}\]
\[k=0.1085\times100=10.85\ \text{N m}^{-1}\approx \boxed{10.8\ \text{N m}^{-1}}\]
(c)(iv) At a mass of 250 g, the extension is 25.0 cm. Therefore,
\[\text{natural length}=38.0-25.0=\boxed{13.0\ \text{cm}}\]
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