PHYS207 Exam 1 20 January, 2016

Transcription

1 PHYS207 Exam 1 20 January, 2016 Work out the problems on the front and back of each page in a clear and logical manner. Please show all your work! Answers without supporting work or justification will not be given credit. This exam consists of 10 problems worth 10 points a piece, although each subpart may not have equal weight. Make sure your exam has all 10 problems in it; do not remove any pages from this exam. You will have exactly 120 minutes. No scientific (graphing) calculators, equation sheets, or any other outside materials are permitted on this exam! Please circle your discussion start time: 12:00 1:00 2:00 Please read the statement below: I hereby agree to follow all directions given by the professor, teaching assistants, and the university. I agree to adhere to all principles outlined in the university s code of conduct. I acknowledge that giving and receiving aid during this exam will constitute a breach of academic integrity, and that I will be disciplined as such. Print your name: Sign your name: Enter today s date:

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4 PHYS207 Exam 1 Page 4 of 24 DO NOT WRITE ON THIS PAGE The following equations might be useful: y = y 0 + v 0 t at2 v = v 0 + at v 2 = v a(y y 0 )

5 PHYS207 Exam 1 Page 5 of (10 points) Suppose a ball B 1 is dropped from rest at an initial height of h above the ground. Simultaneously, another ball, B 2, is thrown directly upwards from the ground. Suppose that the two balls meet at a distance h above the ground. 2 Do not use any numerical values for h or g in the problem! (a) (1 point) In terms of h and g, find the time τ when the two balls collide. (b) (5 points) Using your answer from part (a), determine the initial velocity, v 0, of ball B 2. (c) (2 points) Find the velocity u of ball B 1 when the two balls collide at h 2. (d) (2 points) Find the velocity v of ball B 2 when the two balls collide at h 2.

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7 PHYS207 Exam 1 Page 7 of (10 points) Suppose that a projectile is launched with an initial velocity v 0 = 20 m/s at an angle θ = π 6. Use g = 10 m/s2. For all parts, only substitute numbers into your final expression! DO NOT EVALUATE ANY DECIMALS (a) (1 point) First, draw a diagram illustrating all relevant quantities in parts (b) - (c). Be sure to specify your choice of coordinate system. (b) (3 points) Find the total time of flight τ. (c) (2 points) Using your answer from part (b), find the range R of the projectile. (d) (4 points) Now suppose that a person leaves the launcher at the same time the projectile is launched. If this person has an initial velocity u = 5 3 m/s, what is the acceleration a needed in order to catch the particle when it returns to earth. [Hint: you need to use the answers you found in parts (b) and (c).]

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9 PHYS207 Exam 1 Page 9 of (10 points) Consider the position r(t) of a moving particle given by r(t) = t cos(t)î + t sin(t)ĵ, where r(t) is given in km and t is given in seconds. In m/s and m/s 2, where applicable, calculate: (a) (2 points) the instantaneous velocity v, (b) (3 points) the average velocity u for 0 t 1, (c) (2 points) the instantaneous acceleration a, (d) (3 points) and the average acceleration α for 0 t 1. Note: these are all vector valued. DO NOT EVALUATE ANY TRIG. FUNCTION TO DECIMALS

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11 PHYS207 Exam 1 Page 11 of (10 points) Suppose there is a ring lying on the surface of a table, like what is depicted below, such that there are three people A, B, and C that are pulling on the ring. NOTE: THIS FIGURE IS NOT DRAWN TO SCALE. If the angle θ between B and C is equal to 120 and the magnitude of the forces are F A = 5 3 F C = 5. (a) (3 points) Draw a free body diagram for this system. Be sure to label your choice of coordinate system. There is no coordinate system given in this problem; you must choose your own. (b) (7 points) If we wish to keep the ring from moving, what is the magnitude F B that needs to be exerted by person B in order to keep the ring stationary. Do not substitute any numbers in until the final step. DO NOT EVALUATE TO A DECIMAL AT ANY STEP!

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13 PHYS207 Exam 1 Page 13 of (10 points) Suppose that a particle is shot into the air at an angle θ with an initial velocity v 0. (a) (1 point) Draw a sketch of the situation. Be sure to label your choice of coordinate system. (b) (1 point) What is velocity v of the particle when it is at its maximum height? (c) (1 point) What is the horizontal velocity v f of the particle when it returns to earth. (d) (1 point) What is the maximum (positive) vertical velocity u y? At what time does this occur? (e) (3 point) For which value of θ does the horizontal displacement R become maximal? Be sure to justify your answer.[hint: the trig. identity 2 sin θ cos θ = sin 2θ might be useful.] For parts (f) to (h), consider θ = π 6, π 4, π. [Hint: the following questions do not 3 require long calculations, only basic justifications.] (f) (1 point) For which value of θ does the vertical displacement h become maximal? Be sure to justify your answer. (g) (1 point) For which value of θ does the horizontal velocity u x become maximal? Be sure to justify your answer. (h) (1 point) For which value of θ does the vertical velocity u y become maximal? Be sure to justify your answer.

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15 PHYS207 Exam 1 Page 15 of (10 points) Suppose you are on an alien world where you encounter a creature with N legs and mass m, that hangs upside down from a stationary rod at an angle θ between the fixed and moving legs. The creature s legs are all under the same tension (force). (a) (4 points) What is the ratio of the tension, T, in each leg to the insects weight? (b) (3 points) If the insect straightens out its legs (θ θ + α, α > 0.), does the tension in each tibia increase, decrease, or stay the same? Use the result of part (a) to justify your answer. (c) (3 points) If the insect moves closer to the rod and bends its legs more (θ θ β, β > 0.), does the tension in each tibia increase, decrease, or stay the same? Use the result of part (a) to justify your answer.

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17 PHYS207 Exam 1 Page 17 of (10 points) Consider four blocks that are connected by a rope of negligible mass, as depicted below. All four blocks are being pulled to the right on a horizontal frictionless table by a force T 4 = 50N. Suppose m 1 = 5 kg, m 2 = 10 kg, m 3 = 15 kg, and m 4 = 20 kg. (a) (2 points) What is the acceleration of the entire system? (b) (1 point) What is the acceleration of each individual block. Be sure to justify your answer. (c) (2 points) What is the tension T 1? (d) (2 points) What is the tension T 2? (e) (2 points) What is the tension T 3? (f) (1 point) Draw the free body diagram for the block labeled m 2.

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19 PHYS207 Exam 1 Page 19 of (10 points) Consider two masses m 1 and m 2 connected by a massless rope and ideal pulley, as depicted below. Suppose m 1 is on an incline of angle θ, and the coefficient of friction between m 1 and the incline is µ. Do not use any numerical values in this question (a) (3 points) Derive an expression for acceleration a of the system in terms of g, m 1, m 2, µ and θ. (b) (2 points) Using your answer from part (a), give an expression for the tension T. (c) (2 points) Give a condition on m 2 such that m 1 moves down the ramp. [Hint: this is when a < 0.] (d) (2 points) Give a condition on m 2 such that m 1 moves up the ramp. [Hint: this is when a > 0.] (e) (1 points) Combine your answers from parts (c) and (d) to obtain a condition on m 2 when the system does not accelerate.

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21 PHYS207 Exam 1 Page 21 of (10 points) (a) (5 points) Consider some particle of mass m moving at a velocity v. (i) (1 point) If the velocity is doubled from v to 2v, how much does the kinetic energy change? (ii) (1 point) If the velocity is tripled from v to 3v, how much does the kinetic energy change? (iii) (1 point) If the mass is tripled from m to 3m, how much does the kinetic energy change? (iv) (2 point) Assume that the particle has mass m, and its acceleration is given by a(t) = te t2, and v(0) = 0. (b) (5 points) Find the velocity v of the particle. From this give an expression for the kinetic energy K as a function of time. [Hint: the u-substitution u = t 2 might be useful.] (i) (2 points) Suppose a force F acts over the interval 0 x π, and F (x) = x cos(x 2 ) Find the work done by the force. [Hint: the u-substitution u = x 2 might be useful.] (ii) (3 points) Suppose a force F acts on a particle according to F = 1 + sin(y)ĵ + e zˆk. xî The particle originally starts a coordinates (1, 0, 0) and ends at (e, π, 0). Calculate the work done by F on the particle.

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23 PHYS207 Exam 1 Page 23 of (10 points) An airplane of mass m is flying in a horizontal circle at speed v. Suppose its wings are tilted at an angle θ to the horizontal, as depicted below. Do not use any numerical values in this problem. (a) (1 point) Draw a free-body diagram. Be sure to label your choice of coordinate system. (b) (3 points) What is the radius of the circle in which the plane is flying? Assume that the required lifting force provided is orthogonal to the wing surface. (c) (1 point) How would your solution to (b) change if the mass was doubled from m to 2m? What about if it was increased to 3m? (d) (1 point) How would your solution to (b) change if the velocity was doubled from v to 2v? What about if it was increased to 3v? For parts (e) and (f), you should use a limit based argument. (e) (2 points) What would happen to your solution to part (b) in the limit that θ goes to π 2? (f) (2 points) What would happen to your solution to part(b) in the limit that θ goes to 0?

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