UNIVERSITY OF TORONTO Faculty of Arts and Science DECEMBER 2016 EXAMINATIONS PHY 151H1F Duration - 3 hours Attempt all 10 questions. All questions are worth 5 points. Write your name and student number on every examination book you use. Allowed Aids: calculator (programmable and/or graphing calculators are OK), language dictionary and attached formula sheets (pages 5 and 6). 1. Find the center of mass of the shape shown in Fig.1. Hint: the center of mass of a triangle is located at 1/3 of its height, as measured from the base. w h/2 h w/2 FIG. 1: Shape for Q.1. FIG. 2: Pully and weights for Q.2. 2. A light string, whose ends are attached to two weights m 1 and m 2, is wound over a pulley, of mass M, radius R and moment of inertia 1 2 MR2 (see Fig.2). Assume that the string does not slip on the pulley. Whas the acceleration of m 1? 3. A Ferrari 458 Italia is a type of automobile which has a mass of 1380 kg and its engine has a maximum output power of 312 kw. Ignoring dissipative friction and air resistance, estimate the time required for the car to accelerate from 0 to 100 km/h. FIG. 3: Ferrari 458 Italia (see Q.3). FIG. 4: Figure for question 4. 4. A uniform disc of radius R and mass M is free to rotate abouts centre-axis (see Figure 4). The axis is fixed and horizontal, and the disc rotates in the vertical plane. A small body of mass M is attached to the rim at the highest point above the pivot, and the system is released from rest. Since the system is in an unstable equilibrium, it will soon start to rotate due to some small disturbance. Whas the angular velocity of the system when the body reaches the lowest point, directly below the pivot? (HINT: the moment of inertia of a uniform disc abouts axis is I = 1 2 MR2.) Page 1 of 6
FIG. 5: Potential energy function U(x) for Q.5. 5. Figure 5 shows the total potential energy U (in Joules) as a function of position x (in meters) for a particle of mass m = 1.5 kg which can move in one dimension along the x-axis. At time t = 0 seconds the particle is located at x = 1 and has a velocity v = 2 m/s to the right. There is a kinetic frictional force of constant magnitude F fric = 0.1N acting everywhere. Whas the total distance the particle will travel before it finally PHY151H1F Fall 2016 Final comes to rest? Uncertainty & Python Problem by Jason Harlow 6. (a) Describe the difference between uncertainty and systematic error, and give examples of both. (b) A popular (a) toydescribe is parthe ofdifference a hollow between rubber uncertainty sphere and that systematic popserror, when and inverted give examples andof dropped. both. Is often called a popper. In the Practicals I dropped the inverted popper from a fixed initial and measured the maximum (b) A popular toy is part of a hollow rubber sphere that pops when inverted and dropped. Is often height after the called bounce. a popper. I repeated In the Practicals thisi experiment dropped the inverted 20 times. popper from Thea data fixed initial ranged and from 65 to 95 cm, with an average of 81 cm. measured Using the maximum this information, height after the how bounce. would I repeated you express this experiment the estimated 20 times. The maximum data height, including uncertainty? ranged from 65 to 95 cm, with an average of 81 cm. Using this information, how would you express the estimated maximum height, including uncertainty? (c) Here are some lines of code in a python program. This may not be the entire code, but you can assume that this part (c) Here works; are some islines bugof free, code in and a python doesprogram. what the This programmer may not be the entire intended code, but you to do. can assume that this part works; is bug free, and does what the programmer intended it to do. m = 1.0 y = 2.0 vy = 0 g = 9.8 dt = 0.01 while y > 0 : vy = vy - g*dt y = y + vy*dt u = m*g*y k = 0.5*m*vy**2 e = u + k print("u: ", u, "K: ", k, "E: ", e) i. What physical situation does this code appear to refer to? What are the full names of the i. What physical physical situation quantities doesthat this is code printing appear at the end? to refer to? What are the full names of the physical quantitiesii. thatwhasare printing the first three at the lines end? of output (expressed to 4 significant figures)? iii. Approximately how many times will the while loop execute before stopping? ii. What are the first three lines of output (expressed to 4 significant figures)? iii. Approximately how many times will the while loop execute before stopping? Page 2 of 6
7. A particle of mass m collides with a second particle of mass M (> m) which is initially at rest. In the collision, which may be assumed to be perfectly elastic, the first particle is deflected through a right angle. Show thats speed is reduced by a factor M m M+m by the collision. 8. Tabby s Star is 1500 light years away from earth (measured in the galactic rest frame; assume the star is at rest with respect to earth). In order to send a manned expedition there, a M = 10 5 kg capsule is prepared, with food and oxygen to sustain the crew for 5 years. What kinetic energy must be given to the capsule in order for it to reach its destination before supplies run out? 9. Three perfectly elastic smooth spheres, A, B and C, all of the same diameter and having masses 2 kg, 1 kg and 2 kg respectively are placed, without touching, in a line on a smooth surface with B between A and C. All three are initially at rest; A is then projected toward B with a speed of 27 m/s. Find the final velocities of A, B and C after all impacts. 10. Starting from the Lorentz transformations, derive the relativistic expression for the addition of velocity, i.e. u x = (u x v)/(1 u x v/c 2 ) This is the end of the exam. HAPPY HOLIDAYS EVERYONE! Page 3 of 6
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Mathematical Formulas Formulas You may use these formulas without proof, unless you are specifically requested to derive the result as part of a question. You may detach the formula sheef you wish. Vectors Calculus Scalar Product: F G = F G cos θ = Fx G x + F y G y + F z G z, Vector Product: F G = F G sin θ F G = (F y G z F z G y ) e x + (F z G x F x G z ) e y + (F x G y F y G x ) e z Polynomials: f(t) = t n df dt = ntn 1 Trig functions:(θin radians) f(θ) = cos(θ) df dθ = sin(θ), df f(θ) = sin(θ) dθ = cos(θ) d df Sum of two functions: (f + g) = dt dt + dg Product Rule: d df (fg) = dt dt g + f dg d df dg Chain Rule: f[g(t)] = dt dg dt [ ] t Integration of a polynomial: t n n+1 tf 1 dt = = (n + 1) (n + 1) (tn+1 f t n+1 i ), (n 1). Integral of a sum: [f(t) + g(t)]dt = f(t)dt + g(t)dt, Trigonometric Functions Definitions: sin θ = opposite hypotenuse tan θ = cos θ = opposite adjacent adjacent hypotenuse Useful property (from Pythagoras): sin 2 θ + cos 2 θ = 1 Sum rules: cos(θ + φ) = cos θ cos φ sin θ sin φ, sin(θ + φ) = sin θ cos φ + sin φ cos θ Solutions of Algebraic Equations Quadratic: ax 2 + bx + c = 0 x = b ± b 2 4ac 2a Page 5 of 6
Physics Formulas and constants: Kinematics v = d r d v t t a = v(t) = v() + a(t )dt, r(t) = r( ) + v(t )dt, Newtonian Mechanics Constant acceleration: v(t) = v 0 + a 0 t, r(t) = r 0 + v 0 t + 1 2 a 0t 2 F net = m a = d p p = m v, J = F (t )dt = p, K = 1 2 mv2 Elastic Collision: K and i p i conserved; inelastic collision: i p i conserved E mech = K + U, W = f i F dr, F x = du dx, P = de sys dt = F v ( v constant) Forces F k = µ k N, 0 F s µ s N, F spring = k x Potential Energy U g = mgy, U spring = 1 2 k( x)2 Galilean Relativity (relative motion in the x-direction): t = t, x = (x vt), y = y, z = z, u x = u x v, u y = u y, u z = u z Special Relativity (relative motion in the x-direction; γ = 1/ 1 v 2 /c 2 ): t = γ(t vx/c 2 ), x = γ(x vt), y = y, z = z; u x = u x v 1 u x v/c 2 p = γm 0 u, K = (γ 1)m 0 c 2, s 2 = c 2 t 2 x 2 y 2 z 2 Rotational Motion: s = Rθ, ω = dθ α = dω dθ v = rω = r 2π ω = T r = x e x + y e y = R cos θ e x + R sin θ e y = R e r, v = Rω sin θ e x + Rω cos θ e y = Rω e θ, t t a = Rω 2 e r + Rα e θ, ω(t) = ω( ) + α(t )dt, θ(t) = θ( ) + ω(t )dt, x cm = 1 mn x n = 1 x dm, I = m n rn 2 = r 2 dm, I = I cm + Md 2 M M τ = n r n F n (ext), K rot = 1 2 Iω2, L = r n p n, n d L dt = τ net, Ω = ω ez, L = I Ω (sometimes) Total Marks = 50 Total Pages = 6 Page 6 of 6