Discussion Session 1

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1 Physics 102 Fall 2016 NAME: Discussion Session 1 Math Review and Temperature The goal of Physics is to explain the Universe in terms of equations, and so the ideas of mathematics are central to your success in Physics. You will need to be completely comfortable working with scientific notation (powers of ten), solving algebraic equations (including quadratic equations), manipulating geometric pictures (to find the tangent of some angle, for example), and computing basic derivatives and integrals (using the product and chain rules, if necessary). The first part of this worksheet will ask you to solve some mathematical problems of the type that you are likely to encounter in this Physics course. The second part will discuss some aspects of temperature. 1 Conceptual Questions 1. What is the advantage of using the Kelvin system of temperature, as opposed to the Celsius or Fahrenheit systems? 2 Algebra 1. The equation of an object falling near the surface of Earth without air resistance for a time t under the influence of gravity (with acceleration g = 9.8 m/s 2 ), with an initial upward speed v 0 and initial height h is 2. If y (t) = h + v 0 t 1 2 gt2. Suppose you are standing on a ladder and holding a ball 5 meters above the ground. You throw the ball upwards with a speed of 15 m/s. How long will it take the ball to hit the ground? then show that 1 a + 1 b = 1 c, c = ab a + b. 3. Solve the following system of equations for x and y: 3x + 4y = 32 2x + 5y = 33. 1

2 4. Solve the following system of equations for x and y: 3x + 4y = 32 2x + 5y = 33, this time using matrices. Compare your answer to what you found in question The equation for the (not too fast) velocity of a ball in the presence of air resistance is given to a good approximation by v (t) = v 0 ( 1 e t/τ ), where v 0 is the terminal velocity, and τ is a constant with units of seconds. Determine the initial speed, and how long it would take (in units of τ) for the ball to reach v = v 0 /2. 3 Geometry and Trigonometry 1. For the right triangle in the figure to the right, determine (a) the length of the hypotenuse, (b) the sine, cosine, and tangent of the angle θ, and (c) the angle θ in degrees and radians. Using the values that you found in above, check that the following trig identities hold: (a) cos 2 θ + sin 2 θ = 1 (b) sin (2θ) = 2 sin θ cos θ (c) cos (2θ) = cos 2 θ sin 2 θ (d) cos (θ + 90 ) = sin θ. θ 4 3 2

3 2. Making up some small (x 1) values (and taking angles in radians, such that θ 1), check that the following approximations hold: (a) cos θ 1 (b) sin θ θ (c) e x 1 x (d) (1 + x) x (e) 1 + x x2 (f) (1 + x) 1 1 x. 3. Out fishing one day, you tie a weight to a string to make a pendulum. The end of the string is 25 cm from the middle of the weight. You let the weight hang vertically, and then pull it out (always keeping the string tight) so that it makes an angle of 15 with it s original position, as seen in the figure. (a) How high above it s original hanging height is the weight, now? That is, determine H in centimeters. (b) If you let the weight go, it will swing down and come back up, making the same 15 angle on the other side (neglecting any air resistance). How far from it s starting place does the weight end up on the other side (i.e., what is 2X)? (c) Finally, how far does the weight travel going to the other side (i.e., what is 2D)? H 15 X D γ 4. Consider the collection of triangles seen to the right. What are the angles α, β, γ, δ, and θ? 30 α β θ δ 5. You are riding your bike, which has 62.8 centimeter-diameter wheels, when you get a pebble stuck in the tire. You keep riding for a kilometer, with the pebble stuck firmly in place. How many times did the pebble make an orbit on your tire? 3

4 4 Calculus 1. Find the first and second derivatives of the following functions (a, b, c, and A are constants): (a) x (t) = at 3 + bt 2 + c. (b) x (t) = at 2 + b ln (ct) (c) x (t) = Ae at (d) x (t) = A cos (at). (e) x (t) = A [ cos 2 (at) sin 2 (at) cos(2at) ]. 2. Integrate the following functions: 3. If (a) ax 2 + bx + c dx (b) a cos (bt) dt (c) dx x a (d) 4 0 3x2 dx (e) 2 1 x 1 dx (f) dx x 2 +a 2 and x (t = 0) = 2, then what is x (t = 2)? ẋ (t) = 4t 3, 4. Find all the points, (t, x), where the tangent of the curve x (t) = 6t 2 t is horizontal. 5. Remembering the throwing a ball on a ladder problem from above, the height was given by y (t) = t 4.9t 2, how long will it take the ball to get to it s highest point, and how high is that? 6. Determine the gradient of the following functions (here A, n, k, and ω are all constants), (a) f (x) = Ax n. (b) f (x, y) = A (x 2 + y 2 ). (c) f (x, y) = Ae i( k r ωt). (d) f (x, y, z) = Ae (x2 +y 2 +z 2 ). (e) f (r) = Ar n. 4

5 7. A force( acting ) on a particle in the xy plane at coordinates (x, y) is given by F = (F 0 /r) yî xĵ, where F 0 is a positive constant and r is the distance of the particle from the origin. (a) Show that the magnitude of this force is F 0. (b) Show that the direction of F is perpendicular to r = xî + yĵ. 8. This one is tougher! Suppose that you want to calculate the work done on a particle by the force F = Ar 3ˆr, where A is a constant, moving from the origin, r i = 0î + 0ĵ, to the point r f = î + ĵ. (a) Calculate the work done moving along the path (x, y) = (0, 0) (1, 0) (1, 1). (b) Calculate the work done moving along the path (x, y) = (0, 0) (0, 1) (1, 1). (c) Calculate the work done moving along the path y (x) = x. (d) Calculate the work done moving along the path y (x) = x 2. (e) What do you conclude about the work done along each of these paths? (f) How much work would it take to move from r i, to r f, and then back to r i along a weird path wiggling around all over the place? 9. Find the gradient of the following functions (here, i = 1, and A is a constant) (a) f (x) = 3x. (b) f (x, t) = 4e x2 + 7t 2. (c) f (z) = 6z (d) f (z) = Ae ikz. (e) f (x, y) = Ae k(x2 +y 2 ). 5 Temperature 1. At what temperature is the reading in Fahrenheit and Celsius the same? 2. Convert the following temperatures to their values on the Fahrenheit and Kelvin scales: (a) The freezing point of ice, 0 C. (b) The boiling point of water, 100 C. (c) the sublimation point of dry ice, C, and (d) the human body temperature, 37 C. 5

6 6 Thermal Expansion 1. A concrete highway is built of slabs 12 m long (at 15 C). How wide should the expansion cracks between the slabs be (at 15 C) to prevent buckling if the range of temperature is -30 C to +50 C? 2. To what temperature would you have to heat a brass rod for it to be 1.0% longer than it is at 25 C? 3. A uniform rectangular plate of length l and width w has a coefficient of linear expansion α. Show that, if we neglect very small quantities, the change in area of the plate due to a temperature T is A = 2αlw T. 4. An aluminum sphere is 8.75 cm in diameter. What will be its change in volume if it is heated from 30 to 180 C? 5. (a) Show that the change in density, ρ, of a substance, when the temperature changes by T, is given by ρ = βρ T. (b) What is the fractional change in density of a lead sphere whose temperature decreases from 25 C to -55 C? 6. The pendulum in a grandfather clock is made of brass and keeps perfect time at 17 C. How much time is gained or lost in a year if the clock is kept at 28 C? (Assume the frequency dependence on length for a simple pendulum applies.) 7. A 28.4 kg solid aluminum cylindrical wheel of radius 0.41 m is rotating about its axle in frictionless bearings with angular velocity ω = 32.8 rad/sec. If its temperature is then raised from 20.0 C to 95.0, what is the fractional change in ω? 6

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