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Ch. 36	3	12	17	9	16	30	73
Answers

3.
12. Solve for p. Set M= 4= -q/p. Solve for q in terms of p and substitute into the first equations to get p.
17. (a) To do this problem, remember that for a concave mirror, when the object distance p > f, the image is real (q > 0) and located on the same side of the mirror as the object. When the object distance p< f, the image is virtual (q <0) and on the side of the mirror opposite the object. In this problem, f = 0.5 meters and initially p = 3.00 m. Use: to calculate q initially. Note that : We see that as the ball falls toward the mirror and as p decreases from a value greater than f, q approaches positive infinity. In other words, for p > f, q varies between the initial value that you calculate above and infinity. As the ball continues to fall toward the mirror, for p < f, q varies between negative infinity and zero. Review your algebra, pre-calculus or calculus class notes. Consider q to be a function of p: The mathematical domain of this function is : Physically, we know that p can equal f, but it is helpful to examine the mathematical behavior to predict the physics of the problem i.e. the location of the image as the ball falls. In this problem, the mathematical domain is the interval: To get full credit for this problem, you must do the following: Sketch f(p) on the domain of this problem. Show the vertical asymptote: Hint: This is the vertical line p = f. Show the p-intercept. Hint: This is given by the value of p when q = 0. Show the q-intercept. Hint: This is given by the the value of q when p = 0. It may be helpful to first sketch f(p) on the entire mathematical domain: This will allow you to see the horizontal asymptote, which is the value of q as \|p\| approaches infinity i.e. as p goes to positive or negative infinity. Hint: The horizontal asymptote is the horizontal line q = 1. Review your algebra, pre-calculus or calculus text on how to plot rational functions of the form: (b) If the ball and its image coincide, the image must be real (i.e. on the same side of the mirror as the object): q = p. Thus we have after substituting: Solve for p. You should obtain p = 1.0 m. Show your work. Finally, go back to your notes and/or text from Physics 4A or equivalent and solve for the time it takes a ball to drop from rest through a height of 2.0 m. (Why 2.0 m ? Think about!)
9. Note that f = R/2. Be careful on the sign of R! Remember, this is a convex mirror. (a) Find q and find M from the formula in terms of q and p. (b) Find q and find M from the formula in terms of q and p. (c) Assume the object is upright. Now, is M positive or negative ? Answering this question will tell you if the image is inverted or upright.
16. Before you dive into the mathematics, you should ask yourself whether a concave mirror or a convex mirror represents the Christmas ornament. Think about it ! Note that f = R/2. Be careful on the sign of R! Find q and find M from the formula in terms of q and p. Is the image in front of or behind the mirror?
30. Use: . Solve for f. Find M. The sign of f will tell you whether the lens is converging or diverging.
73. Review your relevant lecture notes and the discussion of two lens systems discussed on page 1160. We assume the two lenses are not in contact with each other. Reread example 36.12 on page 1161 on two converging lenses in combination. Note: Part (c) of this problem deals with two converging lenses as assumed in ex. 36.12. Step 1: Calculate the image distance q₁ for the first lens in the absence of the second lens. Step 2: Calculate the magnification for the first lens in the absence of the second lens. This is given by: M₁ = -q₁/p₁. Step 3: Find the object distance for the second lens. Is the location given by q₁ in step 1 to the right or the left of the second lens ? If the answer to this question is "to the left," then the object distance for the second lens is positive and given by p₂ = 110 m - q₁ . (i.e. the object for the second lens is real.) If the answer to this question is "to the right," then object distance for the second lens is negative and given by p₂ =110 m - q₁. (i.e. the object for the second lens is virtual.) Step 4: Find the final image (in the absence of the first lens) distance q₂ using the formula: Be careful about the sign of f ₂ for a diverging lens ! Step 5: Calculate the magnification for the second lens in the absence of the first lens. This is given by: M₂ = -q₂/p₂. Step 6: Find the overall magnification: (M₁)(M₂). Assuming the initial object is upright, is the final image upright or inverted ? Step 7: Repeat steps 1 to 6 for the case of two diverging lenses. Step 8: Get out of your chair, take a stretch, and relax for a few minutes.