A 5 ft chain is sitting on a table with 2 ft hanging off (3 ft sitting on table). The table is 4 ft high, and the chain slides frictionless off the table. The chain weighs 15 lb. What is the velocity of the chain when it first hits the ground.
To calculate the velocity of the chain when it first hits the ground, we can use the principle of conservation of energy. Here's how we can solve it step by step:
Step 1: Determine the potential energy of the chain when it is completely on the table.
The potential energy (PE) is given by the equation PE = mgh, where m is the mass, g is the acceleration due to gravity (approximately 32.2 ft/s^2), and h is the height.
In this case, the height is 4 ft, and the mass is determined by the weight of the chain. Since the chain weighs 15 lb, we need to convert it to mass in order to use it in the equation:
Weight = mass * gravity
15 lb = mass * 32.2 ft/s^2
mass = 15 lb / 32.2 ft/s^2
Step 2: Calculate the potential energy on the table.
PE = mass * height = (mass) * (4 ft)
Step 3: Determine the kinetic energy when the chain hits the ground.
As the chain slides off the table without friction, its potential energy is converted to kinetic energy. Using the equation for kinetic energy (KE = 0.5 * mass * velocity^2), we can find the velocity of the chain just before it hits the ground.
Step 4: Set the potential energy on the table equal to the kinetic energy just before it hits the ground.
PE = KE
(mass) * (height) = 0.5 * (mass) * (velocity^2)
Step 5: Solve for velocity.
To find the velocity, we rearrange the equation:
velocity^2 = 2 * (height) * gravity
velocity = sqrt(2 * (height) * gravity)
In this case, the height is 3 ft, so:
velocity = sqrt(2 * 3 ft * 32.2 ft/s^2)
Calculating this expression, we find that the velocity of the chain just before it hits the ground is approximately 17.9 ft/s.