Question

A ROD OF LENGTH 3m AND ITS MASS PER UNIT LENGTH IS DIRECTLY PROPORTIONAL TO DISTANCE X FROM ITS END. THE CENTRE OF MASS OF THE ROD OF FROM THAT END WILL BE AT ?

Answer 1

Answer on Question #40896 – Physics – Mechanics

A ROD OF LENGTH 3m AND ITS MASS PER UNIT LENGTH IS DIRECTLY PROPORTIONAL TO DISTANCE X FROM ITS END. THE CENTRE OF MASS OF THE ROD OF FROM THAT END WILL BE AT ?

Solution:

L = 3m – length of the rod;

Mass per unit length is directly proportional to the distance x from its end:

$$\mathrm{dm} = x \mathrm{dx}$$

Mass of the rod:

$$\mathrm{M} = \int_{0}^{\mathrm{L}} \mathrm{dm}$$

Formula for the centre of mass:

$$x_{\mathrm{cm}} = \frac{\int_{0}^{\mathrm{L}} x \cdot \mathrm{dm}}{\mathrm{M}}$$

(1) and (2) in (3):

$$x_{\mathrm{cm}} = \frac{\int_{0}^{\mathrm{L}} x^{2} \mathrm{dx}}{\int_{0}^{\mathrm{L}} x \mathrm{dx}} = \frac{\int_{0}^{\mathrm{L}} x^{2} \mathrm{dx}}{\int_{0}^{\mathrm{L}} x \mathrm{dx}} = \frac{\frac{x^{3}}{3} \left| \frac{L}{0} \right|}{\frac{x^{2}}{2} \left| \frac{L}{0} \right|} = \frac{\frac{L^{3}}{3}}{\frac{L^{2}}{2}} = \frac{L^{3}}{3} \cdot \frac{2}{L^{2}} = \frac{2}{3} \cdot L = \frac{2}{3} \cdot 3 \mathrm{m} = 2 \mathrm{m}$$

Answer: the center of mass of the rod from it's end will be at the distance $2\mathrm{m}$.