The arm of a crane at a construction site is
16.0 m long, and it makes an angle of 13.9
◦
with the horizontal. Assume that the maximum load the crane can handle is limited
by the amount of torque the load produces
around the base of the arm.
What maximum torque can the crane withstand if the maximum load the crane can
handle is 738 N?
Answer in units of N · m.
To find the maximum torque that the crane can withstand, we need to determine the force exerted by the load perpendicular to the arm and multiply it by the length of the arm.
Using trigonometry, we can determine the perpendicular component of the force:
Force perpendicular to the arm = Load * sin(angle)
Substituting the given values:
Force perpendicular to the arm = 738 N * sin(13.9 degrees)
Using a calculator, we find:
Force perpendicular to the arm ≈ 173.03 N
Finally, to find the maximum torque, we multiply the force perpendicular to the arm by the length of the arm:
Maximum torque = Force perpendicular to the arm * length of the arm
Substituting the given values:
Maximum torque = 173.03 N * 16.0 m
Calculating the expression:
Maximum torque ≈ 2768.48 N · m
Therefore, the maximum torque the crane can withstand is approximately 2768.48 N · m.
To find the maximum torque the crane can withstand, we need to use the equation for torque:
Torque = Force x Moment Arm
The force is the maximum load the crane can handle, which is given as 738 N. The moment arm is the perpendicular distance from the line of action of the force to the pivot point, which in this case is the base of the arm.
To find the moment arm, we can use trigonometry. The given information tells us that the arm of the crane is 16.0 m long, and it makes an angle of 13.9 degrees with the horizontal.
The moment arm can be found by multiplying the length of the arm by the sine of the angle:
Moment Arm = Length of the arm x sin(angle)
Moment Arm = 16.0 m x sin(13.9 degrees)
Now we can calculate the maximum torque:
Maximum Torque = Force x Moment Arm
Maximum Torque = 738 N x (16.0 m x sin(13.9 degrees))
Maximum Torque ≈ 1836.7 N · m
Therefore, the maximum torque the crane can withstand is approximately 1836.7 N · m.