An anvil hanging vertically from a long rope in a barn is pulled to the side and raised like a pendulum 1.6 m above its equilibrium position. It then swings to its lowermost point where the rope is cut by a sharp blade. The anvil then has a horizontal velocity with which it sails across the barn and hits the floor, 10.0 m below. How far horizontally along the floor will the anvil land? "
m g h = potential energy at top = U = m (9.8)(1.6)
Ke at bottom = U at top = .5 m u^2
so
1.6*9.8 = .5 u^2
calculate u, the horizontal velocity from that
Now get time in the air from the falling problem
10 = .5 * 9.8 * t^2
so calculate t from that
then
distance horizontal= u t
To determine how far horizontally along the floor the anvil will land, we need to analyze the motion of the anvil during its swing as a pendulum and then during its horizontal motion after the rope is cut.
Let's break down the problem into different steps:
Step 1: Determine the time it takes for the anvil to swing from its highest point to its lowest point.
To do this, we can use the concept of conservation of mechanical energy. At its highest point, the anvil has only potential energy, and at its lowest point, it has only kinetic energy.
The potential energy at the highest point is given by the equation: PE = mgh, where m is the mass of the anvil, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height above the equilibrium position.
The kinetic energy at the lowest point is given by the equation: KE = (1/2)mv^2, where m is the mass of the anvil and v is the velocity at the lowest point.
Using the conservation of mechanical energy, we can equate the potential energy at the highest point to the kinetic energy at the lowest point:
mgh = (1/2)mv^2
Simplifying the equation, we can eliminate the mass of the anvil:
gh = (1/2)v^2
Solving for v, we get:
v = sqrt(2gh)
Step 2: Calculate the time it takes for the anvil to swing from the highest point to the lowest point.
The time it takes for the anvil to swing from the highest point to the lowest point is equal to the period (T) of the pendulum. The period of a simple pendulum is given by the equation: T = 2π * √(L/g), where L is the length of the rope and g is the acceleration due to gravity.
In this case, the length of the rope can be calculated as follows:
L = total length of rope - length raised above equilibrium position
L = 1.6 m
Now, substituting the values into the equation for the period, we get:
T = 2π * √(1.6/9.8)
Step 3: Determine the horizontal velocity of the anvil after the rope is cut.
When the rope is cut, the anvil will continue to move with the same horizontal velocity it had at the lowest point of its swing. Therefore, the horizontal velocity can be calculated as follows:
v_horizontal = v * cos(θ), where v is the velocity at the lowest point (from Step 1) and θ is the angle between the vertical and the direction of motion.
At the lowest point, the anvil is at its equilibrium position, so the angle θ will be 0 degrees.
Step 4: Calculate the horizontal distance traveled by the anvil.
To calculate the horizontal distance traveled by the anvil, we need to know the time it takes for the anvil to hit the floor after the rope is cut. This time can be calculated using the equation: d = v * t, where d is the horizontal distance, v is the horizontal velocity (from Step 3), and t is the time of flight.
The time of flight can be calculated as twice the time it takes for the anvil to swing from the highest point to the lowest point (from Step 2):
t = 2 * T
Now, we can substitute the values into the equation for the horizontal distance:
d = v_horizontal * (2 * T)
Calculating the values will give the final horizontal distance traveled by the anvil.