[br]You will need strong command of mathematics to succeed in physics. These problems are meant to cause you to use that mathematics in ways that are similar to those required by physics. Hint: If you have never been a fan of word problems, please learn (or decide) to like them. After all, that's all that nature has for us! [br][br]The math needed to solve the first set of these problems comes from prior courses and the second set from this chapter. The problems are meant to cause you to think in a way required by physics. Draw pictures to aid visualization. A few of the problems will test your intuition. Have fun learning and extending yourself. [br][br]In this homework and every homework, you need to show all steps of the mathematics. The idea is to demonstrate command of the concepts and mathematics.[br][br]1. The earth rotates 360 degrees on its axis in one sidereal day. This means that the same SIDE (SIDEreal - not the real etymology) faces space after this period of time. This DOES NOT take 24 hours. What takes 24 hours is the time between one side of earth facing the sun and then rotating until the same side faces the sun again. This is called a solar day. Why do you suppose these are different? [br]2. Which should be longer, the sidereal day or the solar day, and why? [br]3. Please assume a circular orbit for the earth, and 365.24 solar days per year to calculate how long (to the nearest minute) a sidereal day should last. [br]4. Consider a rod placed vertically into the ground with exactly one meter above the ground sticking up. Assume you are someplace like Bonneville, UT (famous for auto racing) where the earth will be assumed perfectly level and smooth without hills or valleys or trees to worry about. How far must the stick be from you so that it becomes invisible from a vantage point of being on the ground (face or eyeball at ground level as if you're an ant)? Consider the curvature of the earth in this case. Earth's radius is [math]6.37\times10^6m.[/math] Do the calculation again for an eyeball height of 1.5m above the ground. [br]5. How far will the center of a ball travel if the ball rolls along a level surface without slipping and it rotates exactly once around its center.[br]6. While riding a bicycle, is the top of the wheel moving at the same speed, faster than or slower than the cyclist as measured by a pedestrian standing still.[br]7. How long is the shadow cast on level ground by a 3.0m long vertical stick if the sun is 30 degrees above the horizon? [br]8. What if the vertical stick is on a 5 degree slope with the sun directly uphill and still 30 degrees above the horizon on level ground?   [br]9. What mathematical function f(x) has a slope that is proportional to the value of x? [br]10. What mathematical function f(x) has a curvature proportional to the negative of the function itself? [br]11. What does taking the derivative of a function tell you about the function? Is a derivative a local property in the sense that you can define the derivative of a function f(x) at x? [br]12. What does integrating a function tell you? Is an integral a local property in the sense that you can define the integral of a function f(x) at x? [br]13. Integrate [math]f(x)=3x^2-2[/math] from x=0 to x=2. [br]14. Take the same function’s derivative and evaluate it at x=3.[br]15. If you first differentiate a function and then integrate it, are you going to get the same function back that you started with? [br][br]ANSWERS:[br]1) Because a solar day includes an over-rotation.[br]2) The solar day should be longer.[br]3) 23h 56m[br]4) 3569m (should be an arc segment); 7941m[br]5) A distance equal to the ball's circumference.[br]6) Faster... twice as fast.[br]7) 5.2m[br]8) 6.15m[br]9) a quadratic (parabola)[br]10) a sine or cosine function[br]11) its slope or rate of change. it is local.[br]12) the area under the curve. not local, but over a range.[br]13) 4[br]14) 18[br]15) no. there is always the loss of constant terms upon differentiation that are not recovered upon integration.[br]

Provide full solutions with all math steps shown.[br][br][list=1][*]Why does physics require extensive use of vectors as compared to just scalar quantities? Give a specific example where scalars are insufficient.[/*][*]Given the two complex numbers z[sub]1[/sub] =2+3i and z[sub]2[/sub]=5-3i, what is their sum? What is their difference.[/*][*]Given the complex numbers from problem 2, what is the angle between the x-axis and a line drawn from the origin to the point z[sub]1,[/sub] if plotted in the complex plane?[/*][*]What is the magnitude (also called modulus) of each complex number in problem 2?[/*][*]What is the real part of each complex number in part 2? Imaginary part?[/*][*]Given two vectors [math]\vec{a}=2\hat{x}+3\hat{y}[/math] and [math]\vec{b}=5\hat{x}-3\hat{y},[/math] what is their sum? What is their difference?[/*][*]Given the same two vectors as in problem 6, what is the angle between the x-axis and [math]\vec{a}?[/math][/*][*]What is the magnitude of each vector in problem 6?[/*][*]What is the x-component of each vector in problem 6? y-component?[/*][*]Given the vectors in problem 6, find [math]\hat{a}[/math] and [math]\hat{b}.[/math][/*][*]Rewrite the vectors from problem 6 as [math]\vec{a}=a\hat{a}[/math] and [math]\vec{b}=b\hat{b}.[/math][/*][*]A velocity vector in 3D of a flying rock is given by [math]\vec{v}=20\tfrac{m}{s}\hat{i}-10\tfrac{m}{s}\hat{j}+30\tfrac{m}{s}\hat{k}.[/math] [math]\tfrac{m}{s}[/math] just stands for meters per second, which is the unit for velocity used in the SI system. Speed is the magnitude of the velocity. Find the speed of the rock and its direction of travel. Hint: [math]\vec{v}=v\hat{v}.[/math][/*][*]Given the rock in problem 12, what is the angle between its velocity vector and each (x, y, and z) axis? Do this using direction cosines.[/*][*]Given the vectors in problem 6, find [math]\vec{c}[/math] if [math]\vec{c}=3\vec{a}-2\vec{b}.[/math][/*][*]Find [math]\vec{a}\;\vec{b}.[/math][/*][*]Find [math]\frac{3}{\vec{a}}.[/math][/*][*]Impulse (J) is defined as the change in the momentum of an object, and is given by [math]\vec{J}=\int\vec{F}\;dt[/math] where [math]\vec{F}[/math] is the force (measured in newtons) acting on an object and t is time (in seconds) over which the force acts. What is the change in the momentum of the object in the interval between t=0 and t=5 if the force is given by [math]\vec{F}=2.0t\hat{i}-5.0\hat{j}+\sin(\tfrac{\pi}{5}t)\hat{k}?[/math] In the interval between t=5 and t=10?[/*][*]When it comes to the study of ergonomics (study of interaction between humans and their environments), engineers need to, for instance, consider the comfort of passengers on trains. The sensation of being "jerked" around when throttle is applied suddenly is unpleasant. The amount of "jerking" can be quantified. It is literally called jerk in physics and is equal to the time derivative of the acceleration. Find the jerk associated with an acceleration vector of [math]\vec{a}=4.0t\hat{x}+0.5\sin(2t)\hat{y}.[/math] Then evaluate it at t=1.0. As mentioned, we will be careful to discuss units associated with all such terms soon.[/*][*]What does it tell us about factors on which air drag depends if it is proportional to speed squared?[/*][*]If speed is tripled, how much larger will air drag become for an object? Show the math.[/*][*]What functional form do you expect to describe the motion of a vibrating membrane without damping and why?[/*][*]Suppose a toy boat moves in a pool at at a speed given by v=1.0 meter per second at t=0, and that the boat is subject to viscous damping. The damping on the boat causes the rate of speed loss to be given by the expression [math]\tfrac{dv}{dt}=-2v.[/math] How fast will the boat be traveling after 1 second? 3 seconds? 10 seconds? Use separation of variables to solve this.[/*][*]Find (2.0x10[sup]5[/sup])(3.6x10[sup]4[/sup])/5.3x10[sup]6[/sup] without using parentheses in your calculator. Show exactly how it needs to be typed into the calculator with either how it would need to be entered into MATLAB, or with a photo of it entered on your calculator.[/*][*]Why can't this expression be true: [math]\vec{\tau}=\vec{r}\vec{F}?[/math][/*][*]Why is this not a correct equation: [math]\vec{F}=ma?[/math][/*][*]What's wrong with this expression: [math]K=\tfrac{1}{2}m\vec{v}^2?[/math][/*][*]Why can't this be correct: [math]\vec{a}=7\tfrac{m}{s^2}?[/math] [br][/*][/list][br]ANSWERS: [br]1) Because so many aspects of nature are not adequately described by scalars.[br]2) 7; -3+6i[br]3) 56 degrees[br]4) 3.6; 5.8[br]5) 2, 5; 3, -3[br]6) [math]7\hat{x}[/math]; [math]-3\hat{x}+6\hat{y}[/math][br]7) 56 degrees[br]8) 3.6; 5.8[br]9) 2, 5; 3, -3[br]10) [math]\tfrac{2}{\sqrt{13}}\hat{i}+\tfrac{3}{\sqrt{13}}\hat{j}[/math];[math]\tfrac{5}{\sqrt{34}}\hat{i}-\tfrac{3}{\sqrt{34}}\hat{j}[/math][br]11) [math]\sqrt{13}(\tfrac{2}{\sqrt{13}}\hat{i}+\tfrac{3}{\sqrt{13}}\hat{j})[/math]; [math]\sqrt{34}(\tfrac{5}{\sqrt{34}}\hat{i}-\tfrac{3}{\sqrt{34}}\hat{j})[/math][br]12) 37m/s; [math]\tfrac{20}{37}\hat{i}-\tfrac{10}{37}\hat{j}+\tfrac{30}{37}\hat{k}[/math][br]13) 1.01rad; 1.84rad; 0.64rad[br]14) [math]-4\hat{i}+15\hat{j}[/math][br]15) undefined[br]16) undefined[br]17) [math]25\hat{i}-25\hat{j}+\tfrac{10}{\pi}\hat{k}[/math];[math]75\hat{i}-25\hat{j}-\tfrac{10}{\pi}\hat{k}[/math][br]18) [math]4\hat{x}-0.42\hat{y}[/math][br]19) two independent factors[br]20) nine times[br]21) a sinusoidal one[br]22) 0.14m/s; 0.0025m/s; 2.06x10[sup]-9[/sup]m/s[br]23) 1358.5 should be answer without needing parenthesis[br]24-27) See chapter for rules regarding vector operations.