A sign that weighs 670N is mounted at the end of a 2.00-m long rod of negligible weight. The supporting cable is connected at angle theta=30deg to the center of the rod. What is a) the tension T in the supporting cable, and b) the horizontal and vertical components of the force of the hinge on the rod (Hx and Hy)?
Trying to study for a test. I changed the numbers of the review problem for my sake to see if I could get the actual answer using someone's explanation. (The prof didn't explain it so well, even after asking). So a detailed explanation would be helpful and much appreciated.
To solve this problem, we need to break down the forces acting on the sign and the rod. Let's begin by understanding the forces involved:
1. The weight of the sign acts vertically downward and has a magnitude of 670N.
2. The tension in the supporting cable acts along the cable, pulling the sign upwards and towards the hinge at an angle of 30 degrees with the horizontal.
3. The force of the hinge on the rod has a horizontal component, Hx, and a vertical component, Hy.
Now, let's find the solution step-by-step:
a) Finding the tension T in the supporting cable:
To find the tension in the cable, we need to resolve the forces acting on the sign component-wise. We start by analyzing the vertical forces.
1. Vertical Forces:
In the vertical direction, there are two forces: the weight of the sign (acting downward) and the vertical component of the tension T.
There is no vertical acceleration in this scenario; therefore, the magnitudes of these two forces must be equal.
Weight of the sign = Magnitude of vertical component of T
670N = T * sin(30°)
Solving for T:
T = 670N / sin(30°)
T ≈ 1340N
So, the tension in the supporting cable is approximately 1340N.
b) Finding the horizontal (Hx) and vertical (Hy) components of the force of the hinge on the rod:
To determine the components of the force of the hinge on the rod, we need to analyze the horizontal and vertical equilibrium of the rod.
Since the rod is of negligible weight, the only external forces acting on it are the horizontal and vertical components of the force of the hinge.
1. Horizontal Forces:
In the horizontal direction, there is only one force acting, which is the horizontal component of the force of the hinge (Hx). There is no acceleration in the horizontal direction; therefore, the net force must be zero.
Hx = 0
2. Vertical Forces:
In the vertical direction, there are two forces acting: the weight of the sign (acting downward) and the vertical component of the force of the hinge (Hy).
Hy must counterbalance the weight of the sign to maintain vertical equilibrium.
Weight of the sign = Magnitude of vertical component of the force of the hinge
670N = Hy
So, the horizontal component of the force of the hinge is 0N, and the vertical component is 670N.
To summarize:
a) The tension T in the supporting cable is approximately 1340N.
b) The horizontal component of the force of the hinge on the rod (Hx) is 0N, and the vertical component (Hy) is 670N.