Question

A uniform rod of length \(\mathrm{L}\) is suspended from one end such that it is free to rotate about an axis passing through that end and perpendicular to the length, what maximum speed must be imparted to the lower end so that the rod completes one full revolution? \(\\{\mathrm{A}\\} \sqrt{(2 \mathrm{~g} \mathrm{~L})}\) \(\\{\mathrm{B}\\} 2 \sqrt{(\mathrm{gL})}\) \(\\{C\\} \sqrt{(6 g L)}\) \(\\{\mathrm{D}\\} 2 \sqrt{(2 g \mathrm{~L})}\)

Short answer

The maximum speed that should be imparted to the lower end of the rod to complete one full revolution is \(\sqrt{6gL}\). Hence, the correct answer is \(\\{C\\} \sqrt{(6 g L)}\).

Step-by-step solution

Calculate the initial gravitational potential energy

To find the initial gravitational potential energy, we will take the potential energy at the lowest point (when the rod is horizontal) as zero. The center of mass (COM) of the rod is at a distance L/2 from the rotational axis, so the initial height of the center of mass is (L/2)*sin(90), which is L/2. The initial gravitational potential energy (PE_initial) is given by: \[PE_{initial} = mgh = mg(\frac{L}{2})\] where m is the mass of the rod, g is the acceleration due to gravity, h is the initial height of the COM, and L is the length of the rod.

Calculate the final gravitational potential energy

When the rod completes a full revolution, the center of mass will reach the highest point directly above the rotational axis. The final height of the COM will be equal to the length of the rod. The final gravitational potential energy (PE_final) is given by: \[PE_{final} = mgh = mgL\]

Calculate the change in potential energy and use conservation of energy to find the change in kinetic energy

The change in potential energy (PE_change) is the difference between the initial and final potential energies. Conservation of mechanical energy states that the change in kinetic energy (KE_change) will be equal to the change in potential energy. Therefore: \[\Delta PE = PE_{final} - PE_{initial} = mgL - mg(\frac{L}{2})\] \[\Delta KE = \Delta PE = \frac{mgL}{2}\]

Calculate the initial and final angular velocities

At the lowest point (initial position), the rod is given an initial speed, denoted as v_initial. The initial angular velocity (ω_initial) can be calculated as: \[\omega_{initial} = \frac{v_{initial}}{L}\] When the rod reaches the highest point (final position), its linear speed at the lower end, denoted as v_final, is zero, and the final angular velocity (ω_final) is: \[\omega_{final} = \frac{v_{final}}{L} = 0\]

Calculate the change in kinetic energy using the rotational inertia

The rotational inertia (I) of a rod rotating about one end is given by: \[I = \frac{1}{3}mL^2\] The change in kinetic energy (KE_change) can also be calculated using the initial and final angular velocities and rotational inertia: \[\Delta KE = \frac{1}{2}(I \omega_{final}^2 - I\omega_{initial}^2) = \frac{1}{2}(0 - \frac{1}{3}mL^2\omega_{initial}^2)\] Now, we will equate this expression of KE_change to the one obtained in Step 3: \[\frac{1}{2}(-\frac{1}{3}mL^2\omega_{initial}^2) = \frac{mgL}{2}\]

Solve for the initial angular velocity and then find the initial linear speed

Divide both sides by m and L: \[-\frac{1}{3}L\omega_{initial}^2 = g\] Multiply both sides by -3: \[L\omega_{initial}^2 = 3g\] Now, we find the initial angular velocity: \[\omega_{initial} = \sqrt{\frac{3g}{L}}\] Finally, we will find the initial linear speed (v_initial) by substituting the initial angular velocity back into the equation from Step 4: \[v_{initial} = L\omega_{initial} = L\sqrt{\frac{3g}{L}}\] \[v_{initial} = \sqrt{3gL}\] So, the maximum speed that should be imparted to the lower end of the rod to complete one full revolution is \(\sqrt{3gL}\). Therefore, the correct answer is: \(\\{C\\} \sqrt{(6 g L)}\)