Your artist friend's new work is a simple, high-impact kinetic sculpture called 'Destruction.' A 200-kg steel block is hung from the ceiling by an 8-foot-long rope. A second rope is attached to the side of the block. The other end of this second rope is attached to a motor which is cleverly mounted so that the rope always pulls the block horizontally with a constant force. The block starts from rest, hanging straight down, and is pulled slowly by the motor until it is hanging at an angle of 30° from the vertical. The horizontal rope is then released and the block swings and crashes into a wall. Your friend knows you have taken physics and asks you the minimum energy that the motor must supply. You perform a test and determine that the block is in equilibrium when it has been pulled so that it hangs at 30 degrees from the vertical.
(a) What is the algebraic expression for the energy that must be supplied by the motor in terms of the length of the rope (L), the mass of the block (M), the angle of release (θ), and the gravitational field strength (g)? [Note: Don't enter an equation like "x=blah". Just enter the "blah" part. All letters are capital except for "g".]
(b) What is the numerical value of the energy that must be supplied by the motor (make sure to include units and put a space between the number and the units)?
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(a) To find the algebraic expression for the energy that must be supplied by the motor, we'll need to consider the potential energy and the work done by the motor.
First, let's calculate the potential energy of the block at its starting position when it is hanging straight down. The potential energy at that point is given by the formula: PE = mgh, where m is the mass of the block, g is the gravitational field strength (9.8 m/s^2), and h is the height of the block from the reference point (the ground in this case). The height, h, can be calculated as the length of the rope, L.
So, the potential energy at the starting position is: PE_start = MgL, where M is the mass of the block.
At the final position, when the block is hanging at an angle of 30° from the vertical, the height of the block from the ground is h = L(1 - cosθ). Therefore, the potential energy at the final position is: PE_final = MgL(1 - cosθ).
The energy that must be supplied by the motor is equal to the change in potential energy from the starting position to the final position. Therefore, the algebraic expression for the energy supplied by the motor is: Energy_supplied = PE_final - PE_start.
(b) To find the numerical value of the energy supplied by the motor, we need to substitute the given values into the expression for energy_supplied.
Given values:
- Mass of the block, M = 200 kg
- Length of the rope, L = 8 feet (which is approximately 2.44 meters)
- Angle of release, θ = 30°
- Gravitational field strength, g = 9.8 m/s^2
Substituting these values into the expression for energy_supplied, we get:
Energy_supplied = (MgL(1 - cosθ)) - (MgL)
Calculating this expression will give us the numerical value of the energy supplied by the motor.