1. Consider a rod of mass 1g and length L, held in place against a vertical wall by
a cable at the opposite end as shown in the diagram. A sign of mass 10kg is also
hanging o� the end of the rod furthest from the wall. The angles marked are �=30
�
and �=45
�
.
(a)Find the tension in the cable. (Hint: Is the mass of the rod neg-
ligible here? Then clearly mark your pivot point and start summing your
torques.) If the cable has a breaking point of 1000N, will it hold?
(b) Find the normal force from the wall on the rod and explain
whether or not there must also be friction between the wall and the rod in
this particular problem. If so, give the magnitude and direction of the friction
force required. (Hint: you will need to draw a diagram and sum the forces.
To find the tension in the cable, we need to analyze the forces and torques acting on the rod. Let's break down the problem into smaller steps:
Step 1: Analyzing the forces
Draw a free-body diagram for the rod. We have the weight of the rod acting downward, the tension in the cable acting upward, and the normal force from the wall acting perpendicular to the wall.
Step 2: Choosing a pivot point
To simplify the calculation of torques, we choose a pivot point. In this case, the most convenient choice is the point where the cable attaches to the rod.
Step 3: Summing the torques
With the pivot point chosen, we can calculate the torques. The weight of the rod does not contribute any torque since it acts through the pivot point. The tension in the cable and the normal force both contribute torques.
Step 4: Setting up the equation
The sum of torques must be equal to zero for rotational equilibrium. In this case, we have two torques acting in opposite directions. The tension in the cable creates a clockwise torque, while the normal force creates a counterclockwise torque. Set up the equation with the distances from the pivot point to the respective forces and solve for the tension in the cable.
Step 5: Evaluating the tension and cable strength
Once you have the tension in the cable, compare it to the breaking point of the cable. If the tension is below 1000N, the cable will hold. Otherwise, it will break.
For part (b) of the question, we need to analyze the forces acting on the rod and determine if there is friction between the wall and the rod.
Step 1: Draw a free-body diagram for the rod.
Include the normal force from the wall and the weight of the sign acting downward.
Step 2: Summing the forces
Consider the forces in the vertical direction. The weight of the sign is balanced by the normal force from the wall. There is no vertical acceleration, so the forces must be in equilibrium.
Step 3: Analyzing friction
Friction comes into play when an object is in contact with a surface and there is a tendency for it to slip or slide. In this problem, the rod is held in place by the cable, so there is no tendency for it to slide on the wall. Therefore, there is no requirement for friction in this particular problem.
In summary, to find the tension in the cable, you need to analyze the forces and torques acting on the rod using the method I described. To determine the normal force from the wall and if there is friction, you need to draw a free-body diagram and analyze the forces acting on the rod.