Forces in 2D Legacy Problem #14 Guided Solution
Problem*
A massive light hangs over the table in Jeremy's dining room. The light is supported by four strong chains which make an angle of 75.0° with the horizontal. The force in each chain is 35.8 N.
- Determine the force with which a single chain pulls horizontally and vertically upon the light.
- Determine the mass of the light.
Audio Guided Solution
This is a problem about a massive light hanging at equilibrium and being supported by the force in four different chains. In a problem such as this I am going to organize my information and get a mental picture by constructing a free body diagram. In this diagram I would show five forces. One of the forces is the downforce of gravity pulling up on this light. I am eventually going to have to determine this downforce so that I can find the mass of this light. There are four up and angled forces and they are all angled at the same angle of 75 degrees with the horizontal. So, for simplicity reasons, I could just draw one of them and remember that there are actually four. So, I would draw one force up and to the right. Its force value is 35.8 Newtons. That is the tension in a cable. It makes an angle of 75 degrees with the horizontal. Whenever I have angled forces I am very interested in finding the components of these angled forces. So, what I can do is take the 35.8 Newtons, which is a diagonal force, and resolve it into horizontal and vertical forces. Resolving it into a vertical force means that I am going to have to find the side opposite the 75 degree angle on a force triangle. So, I am going to have to use the sine of 75 and multiply that by the 35.8 Newtons. That gives me a value of 34.6 Newtons, or rather 34.58 Newtons. I should write it down. I am going to use that in subsequent calculations. To find the horizontal pull, I would simply take the 35.8 Newtons, the hypotenuse of a right triangle, and multiply it by the cosine of 75 degrees. That gives me a value of about 9.2657 Newtons. I will round that to 9.3 Newtons. That is my horizontal force and I really do not care much about that force. The question is going to ask me to find the mass of the light. If I want to know the mass of the light, I need to know the downforce of gravity. The horizontal force has nothing to do with that. So I am going to return to the vertical component of the force, which I calculated to be 34.58 Newtons. There are four of these cables pulling up with 34.58 Newtons. So if I take 34.58 Newtons and multiply it by 4, I get 138.3 Newtons. That is the total up-pull on this massive line. If that is the total up-pull and the light hangs in equilibrium, that also is the total down-pull. So the weight of this light, or the F-grab on the light, is 138.32 Newtons down. If I take the 138.32 Newtons and divide by 9.8 Newtons per kilogram, that would give me the mass of the sign. That comes out to be 14.1143 kilograms. I should round that to the proper number of significant digits, 14.1143 kilograms.
Solution
- Fx: 9.3 N and Fy: 34.6 N
- 14.1 kg
Habbits of an Effective Problem Solver
- Read the problem carefully and develop a mental picture of the physical situation. If necessary, sketch a simple diagram of the physical situation to help you visualize it.
- Identify the known and unknown quantities in an organized manner. Equate given values to the symbols used to represent the corresponding quantity - e.g., \(m = \units{1.25}{kg}\), \(µ = 0.459\), \(v_o = \units{0.0}{\unitfrac{m}{s}}\), \(θ = 41.6°\), \(v_f = \colorbox{gray}{Unknown}\).
- Use physics formulas and conceptual reasoning to plot a strategy for solving for the unknown quantity.
- Identify the appropriate formula(s) to use.
- Perform substitutions and algebraic manipulations in order to solve for the unknown quantity.
Read About It!
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