A suspended simple pendulum of length ℓ is making an angle θ with the vertical. On releasing, its velocity at lowest point will be :-

To find the velocity of the simple pendulum at its lowest point, we can use the principle of conservation of mechanical energy.

Step 1: Determine the potential energy at the highest point: At the highest point, the pendulum is at its maximum potential energy. The potential energy is given by the formula PE = mgh, where m is the mass of the pendulum bob, g is the acceleration due to gravity, and h is the height of the pendulum bob from the lowest point. In this case, the height h is equal to ℓ(1 - cosθ), where ℓ is the length of the pendulum and θ is the angle it makes with the vertical.

Step 2: Determine the kinetic energy at the lowest point: At the lowest point, the pendulum is at its maximum kinetic energy. The kinetic energy is given by the formula KE = (1/2)mv^2, where m is the mass of the pendulum bob and v is its velocity.

Step 3: Apply the principle of conservation of mechanical energy: According to the principle of conservation of mechanical energy, the total mechanical energy of the system remains constant. Therefore, the potential energy at the highest point is equal to the kinetic energy at the lowest point.

Step 4: Equate the potential energy and kinetic energy: Setting the potential energy equal to the kinetic energy, we have: mgh = (1/2)mv^2

Step 5: Cancel out the mass: Since the mass of the pendulum bob appears on both sides of the equation, we can cancel it out: gh = (1/2)v^2

Step 6: Solve for the velocity: To find the velocity at the lowest point, we can rearrange the equation: v^2 = 2gh v = √(2gh)

Therefore, the velocity of the simple pendulum at its lowest point is given by v = √(2gh), where g is the acceleration due to gravity and h is the height of the pendulum bob from the lowest point, which is equal to ℓ(1 - cosθ).