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SURNAME: .......................................................... STUDENT NUMBER: INITIALS: ........................................... Signature UNIVERSITY OF CAPE TOWN DEPARTMENT OF PHYSICS PHY1012F Physics A for Engineers FINAL EXAMINATION Thursday 06 June 2019 Time: 120 min + 10 min reading time Internal Examiner: External Examiner: Total marks: 72 Full marks: 70 Dr S.M. Wheaton Dr G. Bosman INSTRUCTIONS: 1. Complete your details at the top of the page. 2. Answer all questions in the spaces provided on the question paper neatly and legibly. 3. Use blank sheets on question paper for all calculations and/or rough work. 4. A list of constants and formulae are available on the last two (2) pages. 5. Prescribed calculators are allowed. 6. Use a pen to complete your answers. 7. Hand in your question paper at the end of the test. 2 Question 1 a. The number 8 ball in a game of pool moves on a horizontal frictionless table. The graphs below indicate 𝑣𝑥 and 𝑣𝑦 components of the ball’s velocity. The ball started from the origin. i. In which direction (angle) is the ball moving at t = 2 s? (Give your answer as an angle from the x-axis.) [2] ii. How far from the origin is the ball at t = 4 s? [4] PHY1012F Exam 2019-drg 3 Question 1…continued b. The number 8 ball of mass 𝑚 = 0.250 kg now strikes the cushion of the pool table at an angle of 𝜃1 = 60.0° at a speed of 𝑣1 = 27.0 m s−1 . It bounces off at an angle of 𝜃2 = 71.0° and a speed of 𝑣2 = 10.0 m s−1. i. What is the magnitude of the change in momentum of the ball? [4] ii. In which direction, with respect to the positive 𝑥-axis does the change in momentum vector point? [1] PHY1012F Exam 2019-drg 4 Question 2 a. Give an example of a particle that has constant kinetic energy but is accelerating. Can a nonaccelerating particle (constant mass) have a changing kinetic energy? If so, give an example. [3] b. Calculate the horizontal force 𝐹⃑ required such that 𝑚1 = 9.00 kg does not slide up or down along the wedge 𝑚2 = 150 kg. All surfaces are frictionless and 𝜃 = 20°. [4] PHY1012F Exam 2019-drg 5 Question 2…continued c. A pendulum of length 𝐿 = 1.0 m with bob of mass 𝑚 = 1.0 kg is released from rest at an angle of 𝜃 = 30° from the vertical. When the pendulum reaches the vertical position, the bob strikes a mass 𝑀 = 3.0 kg that is resting on a frictionless table that has a height ℎ = 0.85 m. i. Calculate the velocity 𝑣⃑ of the bob and the box just after they collide elastically. [5] PHY1012F Exam 2019-drg 6 Question 2…continued ii. Determine how far away from the bottom edge of the table, ∆𝑥, the box will strike the floor. [3] PHY1012F Exam 2019-drg 7 Question 3 a. Your bicycle tires have a radius of 0.33 m. It takes you 14.2 minutes to ride 14 times around a circular track of radius 73 m at a constant speed. i. What is the angular velocity (in rad/s) of the bicycle around the track? [3] ii. What is the angular velocity (in rad/s) of a tire around its axis? [4] PHY1012F Exam 2019-drg 8 Question 3…continued b. A solid sphere of radius 𝑅 and mass 𝑀 is placed at a height ℎ0 on an inclined plane of slope 𝜃. When released, it rolls without slipping to the bottom of the incline. Next a cylinder of the same mass and radius is released on the same incline. If both the cylinder and the 15 sphere reach the same speed at the bottom of the incline, show that ℎ = ℎ0 . [5] 14 PHY1012F Exam 2019-drg 9 Question 4 a. An experiment measures the temperature of a 250 g substance while steadily applying heat to it. The figure below shows the result of the experiment. i. Determine the specific heat capacity (c) of the solid phase. [3] ii. Determine the specific heat capacity (c) of the liquid phase. [3] iii. Determine the heat of fusion (Lf). [2] iv. Determine the heat of vapourisation (Lv). [2] PHY1012F Exam 2019-drg 10 Question 4…continued b. A 200 g piece of copper at temperature 450 K and a 100 g piece of aluminium at temperature 200 K are dropped into an insulated bucket containing water at 280 K. If the thermal equilibrium temperature of the mixture is 282.6 K, what was the mass of the water in the bucket? [4] 𝑐𝐴𝑙 = 900 J/kg K 𝑐𝐶𝑢 = 385 J/kg K 𝑐𝐻2 𝑂 = 4190 J/kg K PHY1012F Exam 2019-drg 11 Question 4…continued c. A monatomic gas follows the process 1 → 2 → 3 shown below. [1 atm = 101325 Pa] i. How much heat is needed for process 1 → 2? [4] ii. How much heat is needed for process 2 → 3? [2] PHY1012F Exam 2019-drg 12 Question 5 a. What is the order of magnitude of the horizontal force exerted on a passenger car travelling at the speed limit (120 km/h) on a freeway when a large truck passes close by going in the opposite direction, also at the freeway speed limit? [4] PHY1012F Exam 2019-drg 13 Question 5…continued b. A 5.00 kg air-filled, sealed, rigid float tank that has a volume of 1.00 m3 is pulled 50.4 m down to the seafloor in order to assist in lifting a sunken object. A diver standing on the seafloor cranks a winch to pull the tank down. i. How much work is required to pull the tank down? (You may safely ignore the mass in the air tank and the mass of the winch rope.) [7] ii. What is the average power required if it takes 10 min to winch down the tank? [3] PHY1012F Exam 2019-drg 14 Physical constants and useful formulae Speed of light Acceleration due to gravity (Earth) Avogadro constant Gas constant Density of air Stefan-Boltzmann constant 𝟏 𝒙(𝒕) = 𝒙𝒐 + 𝒗𝒐𝒙 𝒕 + 𝒂𝒙 𝒕𝟐 𝟐 𝟏 𝒚(𝒕) = 𝒚𝒐 + 𝒗𝒐𝒚 𝒕 + 𝒂𝒚 𝒕𝟐 𝟐 ⃑⃑𝒏𝒆𝒕 = 𝑚𝒂 ⃑⃑ 𝑭 𝝎= 𝒅𝜽 𝒅𝒕 𝒗𝒕 = 𝝎𝒓 3.00 × 108 𝑚/𝑠 9.80 𝑚/𝑠 2 6.022 × 1023 /𝑚𝑜𝑙 8.314 𝐽/𝑚𝑜𝑙 𝐾 1.225 𝑘𝑔/𝑚3 5.67 × 10−8 𝑊/𝑚2 𝐾 4 c g NA R 𝝆𝒂𝒊𝒓 𝒗𝒙 (𝒕) = 𝒗𝒐𝒙 + 𝒂𝒙 𝒕 𝒗𝒚 (𝒕) = 𝒗𝒐𝒚 + 𝒂𝒚 𝒕 𝒇𝒔 (𝒎𝒂𝒙) = 𝝁𝒔 𝑵 𝜶= 𝒅𝝎 𝒅𝒕 𝒔 = 𝒓𝜽 𝒗𝟐 𝒂𝒓 = = 𝝎𝟐 𝒓 𝒓 𝒇𝒌 = 𝝁𝒌 𝑵 𝑻= 𝟐𝝅𝒓 𝟐𝝅 = 𝒗 𝝎 𝒂𝒕 = 𝜶𝒓 𝟏 𝜽𝒇 = 𝜽𝒊 + 𝝎𝒊 ∆𝒕 + 𝜶(∆𝒕)𝟐 𝟐 𝝎𝒇 = 𝝎𝒊 + 𝜶∆𝒕 𝝎𝟐𝒇 = 𝝎𝟐𝒊 + 𝟐𝜶∆𝜽 PHY1012F Exam 2019-drg 15 ⃑⃑ = 𝑭 ⃑⃑𝒇 = 𝒑 ⃑⃑𝒊 𝒑 ⃑⃑ 𝒅𝒑 𝒅𝒕 𝒕𝒇 ⃑⃑ = 𝑱⃑ ∆𝒑 ⃑⃑ (𝒕)𝒅𝒕 𝑱⃑ = ∫ 𝑭 𝒕𝒊 𝒔𝒇 ⃑⃑𝒔 𝒅𝒔 𝑾=∫ 𝑭 ⃑⃑ ∙ ∆𝒓 ⃑⃑ = 𝑭∆𝒓 𝒄𝒐𝒔 𝜽 𝑾=𝑭 𝑷= 𝒔𝒊 𝑼𝒔𝒑 = 𝟏 𝒌(∆𝒔)𝟐 𝟐 𝑭𝒔 = − ∆𝑬𝒔𝒚𝒔 = ∆𝑲 + ∆𝑼 + ∆𝑬𝒕𝒉 = 𝑾𝒆𝒙𝒕 𝑾 ∆𝒕 𝒅𝑼 𝒅𝒔 𝑾𝑵𝑪 = ∆𝑲 + ∆𝑼 𝑲𝒊 + 𝑼𝒊 + 𝑾𝒆𝒙𝒕 = 𝑲𝒇 + 𝑼𝒇 + ∆𝑬𝒕𝒉 ⃑𝝉⃑ = ⃑𝒓⃑ × ⃑𝑭⃑ 𝝉 = 𝒓𝑭 𝒔𝒊𝒏 𝝓 = 𝒓𝑭𝒕 = 𝒅𝑭 𝑰 = ∑ 𝒎𝒊 𝒓𝟐𝒊 = ∫ 𝒓𝟐 𝒅𝒎 𝝉𝒏𝒆𝒕 = 𝑰𝜶 𝒊 ⃑⃑ = 𝒓 ⃑⃑ × 𝒑 ⃑⃑ 𝑳 ⃑⃑ = 𝑰𝝎 ⃑⃑⃑⃑ 𝑳 ⃑⃑𝒏𝒆𝒕 = 𝝉 𝑬(𝒑𝒖𝒓𝒆 𝒓𝒐𝒕𝒂𝒕𝒊𝒐𝒏) = 𝑲𝒓𝒐𝒕 + 𝑼𝑮 = 𝑬(𝒓𝒐𝒍𝒍𝒊𝒏𝒈) = 𝑲𝒓𝒐𝒕 + 𝑲𝒄𝒎 + 𝑼𝑮 = 𝒑= 𝑭 𝑨 ⃑⃑ 𝒅𝑳 𝒅𝒕 𝟏 𝟐 𝑰𝝎 + 𝑴𝒈𝒚𝒄𝒎 𝟐 𝟏 𝟐 𝟏 𝑰𝝎 + 𝑴𝒗𝟐𝒄𝒎 + 𝑴𝒈𝒚𝒄𝒎 𝟐 𝟐 𝒑 = 𝒑𝟎 + 𝝆𝒈𝒉 𝑭𝑩 = 𝝆𝒇 𝑽𝒇 𝒈 𝒗𝟏 𝑨𝟏 = 𝒗𝟐 𝑨𝟐 = 𝑸 𝟏 𝟏 𝒑𝟏 + 𝝆𝒗𝟐𝟏 + 𝝆𝒈𝒚𝟏 = 𝒑𝟐 + 𝝆𝒗𝟐𝟐 + 𝝆𝒈𝒚𝟐 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 𝟐 𝟐 𝑸 = 𝑴𝒄∆𝑻 𝑸 = 𝒏𝑪𝑽 ∆𝑻 𝑸 = ±𝑴𝑳𝒇 𝑸 = 𝒏𝑪𝑷 ∆𝑻 𝒅𝑸 𝒌𝑨 = ∆𝑻 𝒅𝒕 𝑳 𝑸 = ±𝑴𝑳𝒗 𝑾 = −𝒏𝑹𝑻𝒍𝒏(𝑽𝒇 ⁄𝑽𝒊 ) 𝒅𝑸 = 𝒆𝝈𝑨𝑻𝟒 𝒅𝒕 PHY1012F Exam 2019-drg
