A rigid rod holds a restaurant sign horizontally from the side of a building. The force due to gravity acting on the sign is 165N with the angle between the sign and the rod being 50 degrees. Ignoring the weight of the rod what is the magnitude of the tension in the rod?
Fy applied - Fg = 0
Fy applied (y sin theta) = Fg
Fy applied = Fg over sin theta
Fy applied = 165 over sin 50
Fy applied = 215 N
215N
Well, well, well, we've got a sign hanging and a rod in the spotlight! How delightful! Now, let's crunch some numbers and find the magnitude of that tension, shall we?
First things first, we need to break out our trigonometry hats. We have ourselves a right-angled triangle here, with the sign acting as the hypotenuse. The force due to gravity, 165N, will represent the vertical component of the sign's weight.
Now, since we want to find the tension in the rod, we need to take a closer look at the forces involved. The tension in the rod acts as the opposite side of our triangle, and the horizontal component of the sign's weight acts as the adjacent side.
By using some trigonometric magic, specifically the sine function, we can relate the opposite side to the hypotenuse. So, let's dive right in!
Tension = sine(angle) × vertical component
T = sin(50°) × 165N
Now, my math-loving friend, if we plug in the numbers and do some calculations, we'll reveal the magnitude of that tension. And viola!
To find the magnitude of the tension in the rod, we need to analyze the forces acting on the sign. The given force due to gravity is acting vertically downwards, and the tension force in the rod is acting horizontally towards the building.
We can break down the force due to gravity into its horizontal and vertical components. The vertical component can be calculated using the formula:
Vertical force = Force due to gravity * sin(angle between the sign and the rod)
Vertical force = 165N * sin(50 degrees)
Next, we need to determine the horizontal component of the force due to gravity, which is balanced by the tension in the rod. The horizontal component can be calculated using the formula:
Horizontal force = Force due to gravity * cos(angle between the sign and the rod)
Horizontal force = 165N * cos(50 degrees)
Since the sign is in equilibrium, the horizontal force due to gravity is balanced by the tension force in the rod. Therefore, the magnitude of the tension in the rod is equal to the horizontal force due to gravity.
Magnitude of tension in the rod = Horizontal force = 165N * cos(50 degrees)
Calculating this value gives us the magnitude of the tension in the rod.
I do not know because I need to know how far from the building the center of gravity of the sign is and how far from the building the attachment point of the rod to the sign is.
I think
165×sin 50
=126.4 N