one of the factors that limits the reach of a crane is the moment created by the load, measured with the respect to an axis A at the base of the front support. if the boom shown is 30m long, and at an angle of 25 degrees with the horizontal, determine the moment of the 20000 kg load about an axis
To determine the moment of the 20000 kg load about an axis, we need to calculate the perpendicular distance (or lever arm) between the line of action of the load and the axis.
In this case, the axis of rotation is at the base of the front support (axis A). Since we know the length of the boom and the angle it forms with the horizontal, we can use trigonometry to find the perpendicular distance.
Let's apply the sine function to find the vertical component of the boom's length:
sin(angle) = opposite/hypotenuse
sin(25°) = opposite/30m
opposite = sin(25°) * 30m
Next, we can calculate the perpendicular distance from the horizontal line passing through axis A to the line of action of the load. This distance is the horizontal component of the boom's length minus the horizontal component of the length from axis A to the point where the load acts.
Using the cosine function, we can find the horizontal component of the boom's length:
cos(angle) = adjacent/hypotenuse
cos(25°) = adjacent/30m
adjacent = cos(25°) * 30m
Now, let's calculate the horizontal component of the length from axis A to the point where the load acts. Since the load acts vertically downwards from the end of the boom, the horizontal component is simply the distance between the end of the boom and axis A, which is also 30m.
Therefore, the perpendicular distance (lever arm) between the line of action of the load and axis A is:
lever arm = (adjacent - 30m) + 30m
Next, to find the moment (M), we multiply the weight of the load by the lever arm:
M = weight * lever arm
M = 20000 kg * lever arm
Calculating the lever arm using the values we obtained:
M = 20000 kg * ((cos(25°) * 30m) - 30m + 30m)
Simplifying the equation:
M = 20000 kg * (cos(25°) * 30m)
Now, you can calculate the moment (M) by substituting the value of cos(25°) into the equation.