Q. 47 You have a ball of unknown mass,... [FREE SOLUTION]

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Chapter 10: Q. 47 (page 258)

You have a ball of unknown mass, a spring with spring constant $950 N / m$ , and a meter stick. You use various compressions of the spring to launch the ball vertically, then use the meter stick to measure the ball鈥檚 maximum height above the launch point. Your data are as follows:
Use an appropriate graph of the data to determine the ball鈥檚 mass.

Short Answer

Expert verified

The mass is calculated as $6.48 k g .$

Step by step solution

Given information

A ball of unknown mass, a spring with spring constant $950 N / m$ , and a meter stick.

Explanation

The potential energy of a mass h is taken as mgh. Energy stored in the spring with spring constant k when it is compressed by a distance xis $\frac{1}{2} k x^{2} .$

From the calculation below, the spring constant of the spring is $k = 156, 960 N / m .$

A table could be created as follows:

The plot of h vs $x^{2}$ s is given below. The curve is approximately linear. The slope of the curve is approximately $\frac{21}{157} = 0.134$ . The mass is calculated as $6.48 k g$ .

By conservation of energy, we have the following equation

$T h a t i s t h e p l o t o f h v s x^{2} i s l i n e a r . T h e s l o p e o f t h e c u r v e x^{2} = (\frac{2 m g}{k}) h i s \frac{2 m g}{k} .$

The slope can be approximately calculated as

$\frac{2 m g}{k} = \frac{25 - 4}{189 - 32} = \frac{21}{157} m = (\frac{21}{157}) (\frac{k}{2 g}) m = (\frac{21}{157}) (\frac{950}{2 脳 9.81}) m = 6.48 k g$

The energy is conserved. The energy stored in the spring when the spring is compressed is converted to the potential energy of the ball at height h. A linear graph is plotted and its slope is found.