A purple beam is hinged to a wall to hold up a blue sign. The beam has a mass of mb = 6.9 kg and the sign has a mass of ms = 15.3 kg. The length of the beam is L = 2.55 m. The sign is attached at the very end of the beam, but the horizontal wire holding up the beam is attached 2/3 of the way to the end of the beam. The angle the wire makes with the beam is θ = 32.5°.
1) What is the net force the hinge exerts on the beam?
2) The maximum tension the wire can have without breaking is T = 892 N.
What is the maximum mass sign that can be hung from the beam?
To find the net force the hinge exerts on the beam, we can use the concept of torques. A torque is the product of the force applied and the perpendicular distance from the point of rotation to the line of action of the force.
1) To calculate the net force exerted by the hinge on the beam, we need to consider the torques acting on the beam. The torques due to the gravitational forces acting on the beam and the sign will cancel each other out when the system is in equilibrium. We can write the equation for torque as follows:
Στ = τsign − τbeam = 0
where Στ is the net torque, τsign is the torque due to the sign, and τbeam is the torque due to the beam.
The torque due to the sign can be calculated as τsign = Ms * g * Ds, where Ms is the mass of the sign, g is the acceleration due to gravity (approximately 9.8 m/s^2), and Ds is the perpendicular distance from the hinge to the center of the sign.
The torque due to the beam can be calculated as τbeam = Mb * g * Db, where Mb is the mass of the beam, g is the acceleration due to gravity, and Db is the perpendicular distance from the hinge to the center of the beam.
Since the beam is hinged at the wall, Db = 0. Therefore, the torque due to the beam is zero.
Using the given values:
τsign = Ms * g * Ds = 15.3 kg * 9.8 m/s^2 * 2.55 m * (1/3)
Simplifying this equation will give the torque due to the sign.
Next, we set the equation for the net torque to zero:
0 = τsign − τbeam
Substituting the previously calculated values, we can solve for τbeam.
Now, we can calculate the net force exerted by the hinge using the equation:
Net force = τbeam / L
Substituting the calculated value of τbeam and the given values, we can find the net force.
2) To find the maximum mass of the sign that can be hung from the beam without breaking the wire, we need to consider the tension in the wire. The maximum tension the wire can have without breaking is given as T = 892 N.
The tension in the wire is related to the torque due to the sign by the equation:
τsign = T * sin(θ) * L
Simplifying the equation will give the torque due to the sign.
We can then rearrange the equation to solve for the maximum mass of the sign:
Ms = τsign / (g * Ds)
Substituting the previously calculated values and the given values, we can calculate the maximum mass of the sign.