Answer to Question #289709 in Math for Nick

Question

Suppose the length of a spring varies with respect to a weight attached to it. If the length of a spring is 6 cm when a weight of 1 kg is attached and 9 cm when a weight of 3 kg is attached, then which of the following statements is true?

Answer 1

Which of the following statements is true?

a. The original length of the spring(when no weight is attached) is 4.5 cm.

b. The original length of the spring(when no weight is attached) is 5 cm.

c. The length of the spring will be 13.5 cm when a weight of 6 kg is attached.

d. The length of the spring will be 12 cm when a weight of 5 kg is attached.

By the Hooke's Law

$$F=-k\Delta x$$

Let $x_0=$ the original length of the spring(when no weight is attached).

We have

$$m_1g=k(x_1-x_0)$$

$$m_2\cdot g=k(x_2-x_0)$$

Then

$$\dfrac{m_2g}{m_1g}=\dfrac{k(x_2-x_0)}{k(x_1-x_0)}$$

$$m_2(x_1-x_0)=m_1(x_1-x_0)$$

$$x_0=\dfrac{m_2x_1-m_1x_2}{m_2-m_1}$$

$$x_0=\dfrac{3\ kg(0.06\ m)-1\ kg(0.09\ m)}{3\ kg-1\ kg}=0.045\ m$$

$$x_0=4.5\ cm$$

$$k=\dfrac{1\ kg\cdot 9.81m/s^2 }{0.06\ m-0.045\ m}=654 N/m$$

$$x=\dfrac{mg}{k}-x_0$$

$m_3=6\ kg$

$$x_3=\dfrac{6\ kg\cdot 9.81\ m/s^2}{654 N/m}+0.045\ m=0.135\ m$$

$$x_3=13.5\ cm$$

$m_4=5\ kg$

$$x_4=\dfrac{5\ kg\cdot 9.81\ m/s^2}{654 N/m}+0.045\ m=0.12\ m$$

$$x_4=12\ cm$$

a. The statement is True.

b. The statement is False.

c. The statement is True.

d. The statement is True.