Question

When a mass of 40 g is attached to a vertically hanging spring it extends by 0.4 cm. Find :
(i) Force constant of the spring.
(ii) The extension when 100 g weight is attached to it.
(iii) The time period of oscillation of 100 g weight on it.
(iv) The time period and force constant if the spring is cut in three equal parts and 100 g weight is made to oscillate on one part.

Answer 1

Answer on Question #57166, Physics / Mechanics | Relativity

When a mass of $40\,\mathrm{g}$ is attached to a vertically hanging spring it extends by $0.4\,\mathrm{cm}$. Find:

(i) Force constant of the spring.

(ii) The extension when $100\,\mathrm{g}$ weight is attached to it.

(iii) The time period of oscillation of $100\,\mathrm{g}$ weight on it.

(iv) The time period and force constant if the spring is cut in three equal parts and $100\,\mathrm{g}$ weight is made to oscillate on one part.

Solution:

(i) Hooke's Law is a law that shows the relationship between the forces applied to a spring and its elasticity. The relationship is best explained by the equation $F = -kx$. $F$ is force applied to the spring this can be either the strain or stress that acts upon the spring. $x$ is the displacement of the spring with negative value demonstrating that the displacement of the spring when it is stretched. $K$ is the spring constant and details how stiff the spring is.

Since we know that a weight force stretches the spring by $0.4\,\mathrm{cm}$, the spring constant is

$$k = \frac{F}{x} = \frac{mg}{x} = \frac{(0.040\,\mathrm{kg}) \cdot (9.8\,\mathrm{m}/\mathrm{s}^2)}{0.004\,\mathrm{m}} = 98\,\mathrm{N/m}.$$

(ii)

$$x_2 = \frac{F_2}{k} = \frac{m_2 g}{k} = \frac{0.1 \cdot 9.8}{98} = 0.01\,\mathrm{m} = 1\,\mathrm{cm}$$

(iii)

$$T = 2\pi \sqrt{\frac{m}{k}} = 2\pi \sqrt{\frac{0.1}{98}} = 0.2\,\mathrm{s}$$

(iv)

The spring constant will be three times if the spring is cut into three.

$$k_3 = 3k = 3 \cdot 98 = 294\,\mathrm{N/m}$$

Hence,

$$T = 2\pi \sqrt{\frac{m}{3k}} = 2\pi \sqrt{\frac{0.1}{3 \cdot 98}} \approx 0.12\,\mathrm{s}$$

**Answer**: (i) $98\,\mathrm{N/m}$; (ii) $1\,\mathrm{cm}$; (iii) $0.2\,\mathrm{s}$; (iv) $294\,\mathrm{N/m}$ and $0.12\,\mathrm{s}$.

http://www.AssignmentExpert.com/