A uniform rod of length L is pivoted at one end and released from rest. What is
Practice Questions
Q1
A uniform rod of length L is pivoted at one end and released from rest. What is the angular speed when it reaches the vertical position? (2019)
√(3g/L)
√(2g/L)
√(g/L)
√(4g/L)
Questions & Step-by-Step Solutions
A uniform rod of length L is pivoted at one end and released from rest. What is the angular speed when it reaches the vertical position? (2019)
Step 1: Understand that the rod is pivoted at one end and can rotate freely.
Step 2: Recognize that when the rod is released from rest, it has potential energy due to its height.
Step 3: When the rod falls and reaches the vertical position, this potential energy converts into rotational kinetic energy.
Step 4: Use the principle of conservation of energy: Initial potential energy = Final rotational kinetic energy.
Step 5: The potential energy (PE) when the rod is horizontal is given by PE = mgh, where h is the height of the center of mass of the rod.
Step 6: The height h of the center of mass of the rod is L/2 when it is horizontal.
Step 7: Substitute h into the potential energy formula: PE = mg(L/2).
Step 8: The rotational kinetic energy (KE) when the rod is vertical is given by KE = (1/2)Iω², where I is the moment of inertia of the rod about the pivot.
Step 9: The moment of inertia I for a rod pivoted at one end is I = (1/3)mL².
Step 10: Set the potential energy equal to the rotational kinetic energy: mg(L/2) = (1/2)(1/3)mL²ω².
Step 11: Simplify the equation by canceling m and rearranging to solve for ω²: ω² = (3g/L).
Step 12: Take the square root to find the angular speed: ω = √(3g/L).
Conservation of Energy – The principle that the total energy in a closed system remains constant, allowing potential energy to convert into kinetic energy.
Rotational Kinetic Energy – The energy possessed by a rotating object, which is given by the formula KE_rot = (1/2) I ω², where I is the moment of inertia and ω is the angular speed.
Moment of Inertia – A measure of an object's resistance to changes in its rotation, which for a uniform rod pivoted at one end is (1/3) mL².
