The boom of a crane is 20 m long and is inclined 20ᵒ from the horizontal. A bucket of cement with a mass of 3000 kg is suspended from the upper end of the boom. What is the torque acting on the point where the boom is attached to the crane?
A. 5.5 x 105 Nm.
B. 3.5 x 105 Nm.
C. 4.5 x 105 Nm.
D. 2.0 x 105 Nm.
E. 2.0 x 104 Nm
La respuesta es la A
To find the torque acting on the point where the boom is attached to the crane, we need to calculate the moment arm and the force.
The moment arm is the perpendicular distance between the point of rotation (the attachment point) and the line of action of the force. In this case, the moment arm is equal to the length of the boom (20 m).
The force can be calculated by multiplying the mass of the bucket by the acceleration due to gravity (9.8 m/s²). In this case, the force is equal to (3000 kg) × (9.8 m/s²) = 29400 N.
Now we can calculate the torque using the formula:
Torque = force × moment arm
Torque = (29400 N) × (20 m)
Torque = 588000 Nm
Therefore, the torque acting on the point where the boom is attached to the crane is 5.88 x 10^5 Nm.
None of the given options match exactly with the calculated torque. However, the closest option is A. 5.5 x 10^5 Nm.
To find the torque acting on the point where the boom is attached to the crane, we need to calculate the moment of force or torque due to the weight of the bucket.
The torque is given by the expression:
Torque = Force × Lever Arm
The force is the weight of the bucket, and the lever arm is the perpendicular distance from the point of rotation (where the boom is attached) to the line of action of the force.
Given:
Length of the boom (L) = 20 m
Inclination angle (θ) = 20°
Mass of the bucket (m) = 3000 kg
Acceleration due to gravity (g) = 9.8 m/s²
First, let's calculate the force acting on the bucket:
Force = mass × acceleration due to gravity
Force = m × g
Force = 3000 kg × 9.8 m/s²
Force = 29,400 N
Next, we need to calculate the lever arm. The lever arm can be found using trigonometry. In this case, the lever arm is the horizontal component of the boom length:
Lever Arm = Length of the boom × cos(θ)
Lever Arm = 20 m × cos(20°)
Lever Arm ≈ 18.6 m
Now, plug in the values into the torque formula:
Torque = Force × Lever Arm
Torque = 29,400 N × 18.6 m
Torque ≈ 546,840 Nm
Rounding to two significant figures, the torque acting on the point where the boom is attached to the crane is approximately 5.5 × 10^5 Nm.
Thus, the correct option is A. 5.5 × 10^5 Nm.