A 4.00-m-long, 500 kg steel beam extends horizontally from the point where it has been bolted to the framework of a new building under construction. A 70.0 kg construction worker stands at the far end of the beam. What is the magnitude of the torque about the point where the beam is bolted into place?
I searched on this website and found a similar problem, but had an additional question. Is the torque at the center of gravity equal to the torque about the point where the beam was bolted into place?
Here's the work I've done so far.
T = 2.0m * 500kg * 9.81m/s^2 + 4.0m * 70kg * 9.81m/s^2 = 12,557Nm
A beam which is fixed to a wall by bolting or which is extended from inside the building, the free other end of which is freely hanging in the air is called a cantilever. This may make for easier search if you intend to do so.
The torque caused by weights on the beam is called moment, in structural engineering terms (again, easier for searches).
Your formula for the "torque" is correct, namely ΣFD, where F is an applied force on the beam, and D is the distance from the point at which torque is desired.
Note that F carries a sign. If the force "attempts" to turn the beam in the opposite direction, it would be negative.
Here's the cantilever beam:
|
|=========Mg*L/2========mgL
|
Moment (torque)
= ΣFD
= Mg*L/2 + mgL
Well, congratulations on your progress with the problem! As for your additional question, the torque at the center of gravity is not necessarily equal to the torque about the point where the beam is bolted into place.
The torque at a certain point is determined by the force acting on an object and the perpendicular distance from that point to the line of action of the force. In this case, the force acting on the beam is the weight of the construction worker, and the perpendicular distance is the length of the beam.
Since the construction worker stands at the far end of the beam, the torque about the point where the beam is bolted into place will be larger than the torque at the center of gravity. This is because the perpendicular distance is greater at the far end of the beam compared to the center of gravity.
So, in summary, the torque at the center of gravity is not equal to the torque about the point where the beam is bolted into place. Keep up the good work!
To find the torque about the point where the beam is bolted into place, you need to consider the weight of both the beam and the construction worker, as well as their distances from the point of rotation (the bolted point on the beam).
The torque, τ, is given by the formula:
τ = F * r
Where F is the force and r is the distance from the point of rotation.
For the beam:
F = weight of the beam = mass * gravitational acceleration
r = distance of the center of gravity of the beam from the point of rotation
For the construction worker:
F = weight of the construction worker = mass * gravitational acceleration
r = distance of the construction worker from the point of rotation
Assuming the center of gravity of the beam is at its midpoint (2.00 m from the point of rotation) and the construction worker stands at the far end of the beam (4.00 m from the point of rotation), we can calculate the torque.
Weight of the beam (F_beam) = mass_beam * gravitational acceleration
= 500 kg * 9.81 m/s^2
Torque due to the weight of the beam (τ_beam) = F_beam * r_beam
= (500 kg * 9.81 m/s^2) * 2.00 m
Weight of the construction worker (F_worker) = mass_worker * gravitational acceleration
= 70.0 kg * 9.81 m/s^2
Torque due to the weight of the construction worker (τ_worker) = F_worker * r_worker
= (70.0 kg * 9.81 m/s^2) * 4.00 m
Now, to find the total torque about the point where the beam is bolted into place, you can add the torque due to the weight of the beam and the torque due to the weight of the construction worker.
Total torque (τ) = τ_beam + τ_worker
You have correctly calculated the torque for the beam and the construction worker:
τ_beam = (500 kg * 9.81 m/s^2) * 2.00 m
τ_worker = (70.0 kg * 9.81 m/s^2) * 4.00 m
To find the total torque, you can simply add them together:
τ = τ_beam + τ_worker
Substituting the given values into the equation:
τ = (500 kg * 9.81 m/s^2 * 2.00 m) + (70.0 kg * 9.81 m/s^2 * 4.00 m)
Calculating the numerical result gives us:
τ ≈ 12,557 Nm
So, the magnitude of the torque about the point where the beam is bolted into place is approximately 12,557 Nm.
Regarding your additional question, the torque at the center of gravity is not necessarily equal to the torque about the point where the beam is bolted into place. The torque depends on the distance from the point of rotation, so if the center of gravity is not at the point of rotation, the torque will be different. In this case, the construction worker's weight creates an additional torque contribution.
To find the magnitude of the torque about the point where the beam is bolted into place, you need to consider the forces acting on the beam.
First, let's calculate the torque created by the weight of the steel beam. The weight acts downward at the center of the beam, which is at a distance of 2.0 m from the point where it is bolted into place. The formula for torque is given by:
Torque = Force × Distance
The force is the weight of the steel beam, which is equal to the mass multiplied by the gravitational acceleration:
Force_1 = 500 kg × 9.81 m/s²
The distance is the distance from the point of rotation (the point where the beam is bolted into place) to the center of gravity of the beam, which is 2.0 m:
Distance_1 = 2.0 m
So, the torque created by the weight of the steel beam is given by:
Torque_1 = Force_1 × Distance_1
Next, let's calculate the torque created by the weight of the construction worker. The weight acts downward at the far end of the beam, which is at a distance of 4.0 m from the point where it is bolted into place. The formula for torque is the same as before:
Torque = Force × Distance
The force is the weight of the construction worker, which is equal to the mass multiplied by the gravitational acceleration:
Force_2 = 70 kg × 9.81 m/s²
The distance is the distance from the point of rotation to the location of the construction worker at the far end of the beam, which is 4.0 m:
Distance_2 = 4.0 m
So, the torque created by the weight of the construction worker is given by:
Torque_2 = Force_2 × Distance_2
To find the total torque about the point where the beam is bolted into place, simply add the two torques:
Total Torque = Torque_1 + Torque_2
Now let's calculate the torques:
Force_1 = 500 kg × 9.81 m/s² = 4905 N
Force_2 = 70 kg × 9.81 m/s² = 686.7 N
Torque_1 = Force_1 × Distance_1 = 4905 N × 2.0 m = 9810 Nm
Torque_2 = Force_2 × Distance_2 = 686.7 N × 4.0 m = 2746.8 Nm
Total Torque = Torque_1 + Torque_2
Total Torque = 9810 Nm + 2746.8 Nm = 12,556.8 Nm
Therefore, the magnitude of the torque about the point where the beam is bolted into place is 12,556.8 Nm.
Now, moving on to your additional question:
Is the torque at the center of gravity equal to the torque about the point where the beam was bolted into place?
No, the torque at the center of gravity is not necessarily equal to the torque about the point where the beam is bolted into place. The center of gravity is the point where the weight of an object is concentrated, and the torque at that point is typically zero. In this scenario, the torque at the center of gravity of the beam is zero because the line of action of the weight passes through it.
However, the torque about the point where the beam is bolted into place is not zero. It is determined by the forces acting on the beam and their respective distances from the point of rotation. As we calculated earlier, the torque about the point where the beam is bolted into place is 12,556.8 Nm.