Equlibrium and Gravitational Force

[amie]

Homework Statement

In the figure below, a uniform beam of length 12.0 m is supported by a horizontal cable and a hinge at angle θ = 70°. The tension in the cable is 500 N.
(a) What is the gravitational force on the beam, in unit-vector notation?

(b) What is the force on the beam from the hinge, in unit-vector notation?

http://www.webassign.net/hrw/12-73.gif

Homework Equations

Torque = r x f

The Attempt at a Solution

The system is in equilibrium and so I assumed that the two should be the opposites of each other. My real problem, however is I have no idea how to calculate the Torque or the gravitational force without knowing the weight on the beam.

 

[PhanthomJay]

Homework Statement

In the figure below, a uniform beam of length 12.0 m is supported by a horizontal cable and a hinge at angle θ = 70°. The tension in the cable is 500 N.
(a) What is the gravitational force on the beam, in unit-vector notation?

(b) What is the force on the beam from the hinge, in unit-vector notation?

http://www.webassign.net/hrw/12-73.gif

Homework Equations

Torque = r x f

The Attempt at a Solution

The system is in equilibrium and so I assumed that the two should be the opposites of each other.

not true, sum moments to get reaction force

My real problem, however is I have no idea how to calculate the Torque or the gravitational force without knowing the weight on the beam.

But you are given the tension in the cable.

 

[kerimek]

I would like to first clarify that the term "equilibrium" in this context refers to a state where all forces acting on an object are balanced, resulting in a net force of zero. In this case, the beam is in rotational equilibrium, meaning that the sum of all the torques acting on it is also equal to zero.

To answer the first part of the question, we can use the equation for torque, which is the cross product of the position vector and the force vector. In this scenario, the gravitational force acts on the beam at its center of mass, which is at a distance of 6.0 m from the hinge. Therefore, the torque due to gravity can be calculated as:

τ = r x F = (6.0 m) x (m x g) = 6.0 m x (12.0 kg x 9.8 m/s^2) = 705.6 Nm

Since the beam is in equilibrium, the torque due to the gravitational force must be balanced by the torque due to the tension in the cable. Therefore, the tension force can be calculated as:

τ = r x F = (6.0 m) x (F) = 6.0 m x (500 N) = 3000 Nm

To answer the second part of the question, the force from the hinge can be calculated using the same equation, but with a different position vector. In this case, the force from the hinge acts at the hinge itself, which is at a distance of 12.0 m from the center of mass. Therefore, the force from the hinge can be calculated as:

τ = r x F = (12.0 m) x (F) = 12.0 m x (F) = 0 Nm

Since the torque due to the hinge force is zero, the force from the hinge must also be zero. This makes sense, as the hinge only acts as a pivot point and does not exert any horizontal force on the beam.

In conclusion, the gravitational force on the beam can be calculated using the equation for torque, taking into account the distance of the center of mass from the hinge. The force from the hinge, on the other hand, is zero as it only acts as a pivot point.