A rod of length L, whose lower end is resting along the horizontal plane starts to topple from a vertical position . What will be the velocity of the end when it hits the ground?
Answers
When a rod hits the ground, the velocity of the end will be √3gL.
Explanation:
=> When upper end of the rod hits the ground, the distance through which centre of gravity of the rod falls =L/2.
Mass of the rod = M.
Thus, Loss of potantial energy =Mg *L/2
=> According to the law of energy conservation,
Initial PE = Final KE
mg*L/2 = 1/2 * Iω² ...(1)
=> Moment of inertia can be given as, I:
I = mL²/12
=> By putting the value of I in eq(1), we get
mg* L/2 = 1 * mL² * ω² / 2 * 12
ω² = m*g * L * 12 * 2 / 2 * m *L²
ω² = 12g/L
ω = √12g/L
=> Linear velocity at the end of the rod will be:
v = L*ω
= L/2 * √12g/L
= L/2 * √3*4 gL/L²
= L/2 * 2/L * √3gL
= √3gL
Thus, the velocity of the end will be √3gL.
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