At one point during its swing, a wrecking ball exerts a tension force of
FT = 9825 N on its cable, which makes an angle of α = 25.0° with the horizontal. The crane's 9.00-m-long boom is at an angle of θ = 60.0° with the horizontal. What is the torque exerted by the wrecking ball on the crane about an axis perpendicular to the page and passing through point P shown in the figure below? (Enter the magnitude
To find the torque exerted by the wrecking ball on the crane, we need to use the formula for torque:
Torque = force × perpendicular distance
First, let's label the given values in the problem:
- Tension force (FT) = 9825 N
- Angle of the tension force (α) = 25.0°
- Length of the crane's boom (L) = 9.00 m
- Angle of the crane's boom (θ) = 60.0°
To calculate the torque, we need to find the perpendicular distance from the point P to the line of action of the force. In this case, the line of action is the direction of the tension force.
The perpendicular distance can be calculated using trigonometry. Since we have the length of the boom and the angle θ, we can find the horizontal component of the boom (L⋅cos(θ)) and the vertical component of the boom (L⋅sin(θ)). Then, using the vertical component and the angle α, we can find the perpendicular distance (d) from the line of action of the force to point P.
Perpendicular Distance (d) = vertical component of the boom × sin(α)
Now that we have the perpendicular distance, we can calculate the torque:
Torque = Tension force × perpendicular distance
Torque = FT × d
Substituting the given values into the equation, we can calculate the torque.