Question:

A body of mass 'm' is projected with a speed 'u' making an angle of 4545^\circ with the ground. The angular momentum of the body about the point of projection, at the highest point, is expressed as 2mu3Xg\frac{\sqrt{2} \, m u^3}{X g}. The value of 'X' is ______.

Updated On: Nov 19, 2024
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Correct Answer: 8

Solution and Explanation

At the highest point, the vertical component of the velocity becomes zero, and only the horizontal component ux=ucos45=u2 u_x = u \cos 45^\circ = \frac{u}{\sqrt{2}} remains.

The maximum height h h reached by the body is given by:

h=(usin45)22g=(u2)22g=u24g. h = \frac{(u \sin 45^\circ)^2}{2g} = \frac{\left(\frac{u}{\sqrt{2}}\right)^2}{2g} = \frac{u^2}{4g}.

The angular momentum L L about the point of projection at the highest point is:

L=muxh=mu2u24g=2mu38g. L = m \cdot u_x \cdot h = m \cdot \frac{u}{\sqrt{2}} \cdot \frac{u^2}{4g} = \frac{\sqrt{2}mu^3}{8g}.

Thus, the value of X X is:

X=8. X = 8.
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