Solving Problems With Newton’s Laws

If object A exerts a force on object B, then object B exerts and equal and opposite force on object A. Examples: ( identify all the action- reaction pairs I want to remind you again of Newton’s third law… 4.5 Newton’s Third Law of Motion

Example: ( Let’s just look at the “rope” connecting the two masses. We will learn how to solve this problem soon. I’ve introduced the concept of “tension” in this example. Read about on page 92 of your text.

Newton’s third law gives us one new OSE: How to remember the order of the subscripts: we will be particularly interested on the forces ON an object. Forces on come first. Therefore, you may write (if you wish)

Normal force example: the table exerts an upward force on the lamp. Normal force example: the ramp exerts a force on the block. Normal force example: the table exerts an upward force on the lamp. The force is normal to the surface of the table. Normal force example: the ramp exerts a force on the block. The force is normal to the surface of the ramp. N N The Normal Force I also want to remind you about…

4.7 Solving Problems with Newton’s Laws: Vector Forces and Free-Body Diagrams Remember Newton’s second law: I could also write this The acceleration of any object is proportional to the vector sum of all the forces on the object. In all of our problems, we will consider the different vector components one at a time, rather than all at once.

As an example of a force problem, I’ll work the inclined plane problem I showed you a few slides back. I’ll follow along with the Litany for Force Problems. You should work out other examples in section 4.7 (and your homework, of course) using the Litany. I’ll introduce several important concepts in this example, so stop me if something doesn’t make sense. Example: a wedge makes an angle θ with the horizontal. The surface of the wedge is frictionless. A frictionless pulley is attached at the raised end. Two masses, M and m, are connected by a massless cord, as shown. Calculate the acceleration of the system. M m

M m Step 1: draw a basic representative sketch. We have a sketch, but we should illustrate the forces on the sketch. Forces on mass M are the tension in the cord, its weight, and the normal force due to the wedge. You could write W=Mg beside W if you wanted. Forces on mass m are the tension in the cord, and its weight. Note: same T as on M, different direction. You could write w=mg beside W if you wanted. T N W w T

Hint: exaggerate the shape of your inclined plane so you don’t have to use trig to figure out which angle goes where. In a minute we are going to define axes and show force components. Before I do that… θ W θ ?

I will do the rest of the work on the blackboard, listing the steps here. Step 2. Draw a free-body diagram with all forces shown as vectors originating from the labeled point mass you are considering. Show an appropriate acceleration vector next to the point mass (or write "a = 0" if the acceleration is zero). This can be superimposed on the sketch if it can be done clearly, with the vectors drawn darker than the lines of the sketch. Each of these vectors must be distinctly and legibly drawn.

Step 3. Label each vector with an appropriate symbol. Use subscripts if necessary, e.g. T 1 and T 2, to distinguish between two different vectors of one type quantity. Step 4. Lightly draw an appropriate coordinate axis- system near or on the free-body diagram, with an arrow at one end of each line indicating the positive direction of that axis. Choose the orientation of the axes to fit the problem. If the direction of acceleration is constant and known, it is almost always best to choose one of the axes to be in the direction of (or at least parallel to) the acceleration vector. Otherwise, choose axis orientation to handle force components conveniently.

Step 5. Lightly draw in vector projections of all force vectors that are not parallel to a coordinate axis. Make sure there are arrows at the end of these vectors to indicate their direction. Step 6. As an initial mathematical step, you MUST begin with an appropriate Official Starting Equation. All subsequent steps must mathematically follow from this beginning point and reference to your diagram. Step 7. Write out the sum of force components explicitly, with the number of terms matching the number of force vectors in your free body diagram. Sometimes you may want to leave the component as a symbol (maybe you don’t know its direction), but make sure it has the proper axis-subscript, e.g., x in T 1x.

Step 8. Solve for the desired quantity algebraically. If you use a symbol in an equation, it must appear in your diagram. If you don’t know the direction of a vector, e.g., acceleration in this example, it is best to leave it in component form. Why? If you choose to skip steps, you dramatically increase your chances of making mistakes. Hold off on the substitution of numerical values for the symbols until the end of the solution. Draw a box around your final answer.