A pulley system os being used to raise a load. The rope begins at the ceilings, goes down, round a pulley on the load, up to a pulley on the ceiling, back down and around a second pulley on the load, and up to a sheave (a motor driven pulley) on the ceiling. The rope is vertical between the pulleys. The sheave has diameter 90mm and angular 47 rev per min. What is the speed with the load is being raised?
The sheave brings in rope at a rate R*w = .045 m*2*pi*47/60 rad/s = 0.2215 m/s
Since there are four rope strands betqween load and ceiling, the load is raised at 1/4 that rate, 0.0554 m/s
To determine the speed at which the load is being raised, we need to consider the rotational speed of the sheave and the relationship between the angular speed and the linear speed of a rotating object.
The angular speed of the sheave is given as 47 revolutions per minute. To convert this to radians per second, we need to multiply by 2π/60:
Angular speed = (47 rev/min) * (2π radians/60 min)
Angular speed = 47 * (2π/60) radians per second
Angular speed ≈ 4.93 radians per second
Now, we have the angular speed of the sheave. To calculate the linear speed, we need to use the formula:
Linear speed = Angular speed * Radius
The radius of the sheave is half its diameter, so:
Radius = 90mm / 2 = 45mm = 0.045m
Plugging in the values:
Linear speed = 4.93 radians/second * 0.045m
Linear speed ≈ 0.222 m/s
Therefore, the speed at which the load is being raised is approximately 0.222 meters per second.