Forces in 2D Legacy Problem #4 Guided Solution
Problem*
At one moment during a walk around the block, there are four forces exerted upon Fido - a 10.0 kg dog. The forces are:
- Fapp = 67.0 N at 30.0° above the horizontal (rightward and upward)
- Fnorm = 64.5 N, up
- Ffrict = 27.6 N, left
- Fgrav = 98 N, down
Resolve the applied force (Fapp) into horizontal and vertical components, then add the forces up as vectors to determine the net force.
Audio Guided Solution
The net force is the vector sum of all the individual forces which act upon an object. That means you have to take all the forces and add them up, treating them as vectors, with an up, down, and a right, left direction. Now if you have a force that's neither going up, nor going right, nor going down, nor going left, strictly, but instead is going a combination of the two directions, then the only method of solving the problem is to take that nasty force and to resolve it into its components. That is to resolve it into its effect in the horizontal and the vertical direction. And that's what we have here with the 67 newton force at a 30 degree angle above the horizontal. What you'll do with that force is you'll resolve it into a horizontal and a vertical component. To find its horizontal component, you would have to take 67 newtons and multiply by the cosine of 30 degrees. Cosine because the 30 degrees is the angle that that force makes with the horizontal line. And as such, it's the horizontal side, which is adjacent to the 30 degrees. To find its vertical component, you'd have to go 67 newtons times the sine of 30 degrees. Sine of 30 degrees because 30 degrees is the angle that the 67 newtons makes with the horizontal, and so it would be the vertical side, which would be opposite 30 degrees on the right triangle. Now when you do that, 67 newtons times the cosine of 30 degrees, you get 58 newtons to the right. And when you go 67 newtons times the sine of 30 degrees, you get 33.5 newtons up. Now what you've done is you've taken four forces, three of which have been very cooperative all along, and one which has been nasty, and you have substituted for the nasty force. Two forces, one which is up, and the other which is to the right. Now you have five forces and you can add them all up, and as you do, you have to add the ups and downs together, and the rights and lefts together. Adding the ups and downs together is a matter of taking the two ups, 64.5 newtons of normal force and 33.5 newtons of applied force vertically, and adding them up to get 98 newtons, and then add to that the 98 newtons down. As you can see, the ups and downs actually balance each other out. What you're left with here is 58 newtons to the right to be added to 27.6 newtons to the left. If you add those two forces up as vectors, you get 30.4 newtons to the right, and that would be the answer to what is the F net acting up on the object.
Solution
Fapp-horizontal: 58.0 N, right
Fapp-vertical: 33.5 N, up
Fnet: 30.4 N, right
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!
Get more information on the topic of Forces in 2D at The Physics Classroom Tutorial.