A uniform rod of length L and mass M is pivoted at one end and released from res
Practice Questions
Q1
A uniform rod of length L and mass M is pivoted at one end and released from rest. What is the angular speed of the rod just before it hits the ground? (2019)
√(3g/L)
√(2g/L)
√(g/L)
√(4g/L)
Questions & Step-by-Step Solutions
A uniform rod of length L and mass M is pivoted at one end and released from rest. What is the angular speed of the rod just before it hits the ground? (2019)
Step 1: Understand the problem. We have a uniform rod of length L and mass M that is pivoted at one end and falls due to gravity.
Step 2: Identify the energy types involved. Initially, the rod has potential energy (PE) when it is held upright. As it falls, this potential energy converts into rotational kinetic energy (KE).
Step 3: Write the formula for potential energy. The potential energy (PE) at the top is given by PE = M * g * h, where h is the height of the center of mass of the rod. For a rod of length L, the center of mass is at L/2, so h = L/2. Thus, PE = M * g * (L/2).
Step 4: Write the formula for rotational kinetic energy. The rotational kinetic energy (KE) when the rod is about to hit the ground is given by KE = (1/2) * I * ω², where I is the moment of inertia of the rod about the pivot point and ω is the angular speed.
Step 5: Calculate the moment of inertia (I) of the rod. For a rod pivoted at one end, I = (1/3) * M * L².
Step 6: Set the potential energy equal to the rotational kinetic energy. M * g * (L/2) = (1/2) * (1/3) * M * L² * ω².
Step 7: Simplify the equation. Cancel M from both sides and rearrange to find ω²: g * (L/2) = (1/6) * L² * ω².
Step 8: Solve for ω². Multiply both sides by 6: 3gL = L² * ω². Then divide by L²: ω² = 3g/L.
Step 9: Take the square root to find ω. Thus, 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 Kinematics – The study of the motion of objects that rotate, including the relationships between angular displacement, angular velocity, and angular acceleration.
Moment of Inertia – A measure of an object's resistance to changes in its rotation, which is crucial for calculating rotational kinetic energy.
