Dr. Gundersen Phy 205DJ Test 2 22 March 2010

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1 Signature: Idnumber: Name: Do only four out of the five problems. The first problem consists of five multiple choice questions. If you do more only your FIRST four answered problems will be graded. Clearly cross out the page and the numbered box of the problem omitted. Do not write in the other boxes. TO GET PARTIAL CREDIT IN PROBLEMS 2-5 YOU MUST SHOW GOOD WORK CHECK DISCUSSION SECTION ATTENDED: [ ] Dr. Gundersen 2O, 9:30-10:20 a.m. [ ] Dr. Nepomechie 2P, 11:00-11:50a.m. [ ] Dr. Alvarez 2Q, 12:30-1:20 p.m. [ ] Dr. Barnes 2R, 2:00-2:50 p.m. [ ] Mr. Perez-Veitia 2S, 3:30-4:20 p.m. a = a x î + a y ĵ, a = a = b = bx î + b y ĵ + b zˆk, a 2 x + a 2 y, Vector relations: θ = tan 1 a y a x, a x = acos θ, a y = asin θ b = b = b 2 x + b 2 y + b 2 z, ˆb = b/b, vac = v AB + v BC a b = a b cos θ = a x b x + a y b y + a z b z For constant acceleration: x = x 0 + v 0 t at2, v 2 = v a(x x 0 ), x = x (v 0 + v)t, x = x 0 + vt 1 2 at2 v = v 0 + at, v av = 1 2 (v 0 + v) = v at2, r(t) = r 0 + v 0 t at2, v(t) = v 0 + at For nonconstant acceleration: v = d r dt, v av = r 1 r t1 0, x 1 x 0 = v x (t)dt t 1 t 0 t 0 a = d v dt, a av = v 1 v t1 0, v x1 v x0 = a x (t)dt t 1 t 0 t 0

2 Forces: F net = m a, F BA = F AB, F g = mg, 0 f s µ s F N, f k = µ k F N F = mv2 R ˆr, Fs = k d, F du = dx î du dy ĵ du dz ˆk, F du(r) = ˆr dr Uniform Circular Motion: r(t) = r(cos θî + sinθĵ) = rˆr, v(t) = ωr( sin θî + cos θĵ) = ωrˆθ, θ(t) = ωt + θ 0 a(t) = ω 2 r(cos θî + sinθĵ) = ω2 rˆr = v2 dθ ˆr, ω = r dt Nonuniform Circular Motion: a tot = a rad + a tan = v2 r ˆr + αrˆθ, α = dω dt = d2 θ dt 2 Work, Kinetic Energy and Power: = 2πf = 2π/T = v/r W = F x x, K = 1 2 mv2, W = 1 2 mv2 1 2 mv2 0 = K, P = dw dt = F v, W g = mgdcosφ W = F x x + F y y + F z z = F d = F d cos φ, W = rb r A Potential Energy and Energy Conservation: F d r, Ws = 1 2 kx2 i 1 2 kx2 f xf U = F(x)dx = W = K, U(y) = mgy, U(x) = 1 x i 2 kx2, E th = f k d P = de dt, E mech = K + U, W = E mech + E th + E int

3 [1.] This problem has five multiple choice questions. Circle the best answers. [1A.] A constant force F = (4.00 N)î+(3.00 N)ĵ acts on a 2.00 kg object as it moves in a straight line from an initial position given by r 1 = (1.00 m)î + (1.00 m)ĵ to a final position given by r 2 = (4.00 m)î (1.00 m)ĵ. Determine the work done by the force during this motion. [a] 7.00 J [b] J [c] 13.0 J [d] 2.00 J [e] 6.00 J [1B.] What is the angle between F and the displacement vector d = r 2 r 1 in the above problem? [a] θ = cos 1 5 ( 6 ) 13 [b] θ = cos 1 6 ( 5 ) 13 [c] θ = cos 1 ( 6 65 ) [d] θ = cos 1 ( 2 15 ) [e] θ = cos 1 ( 6 7 ) [1C.] A force on a particle depends on position such that F(x) = (3 N/m 2 )x 2 + (6.00 N/m)x. The particle is constrained to move along the x-axis. What work is done by this force as the particle moves from x = 0.00 m to x = 2.00 m? [a] J [b] 10.0 J [c] 20.0 J [d] 48.0 J [e] 24 J [1D.] A ball of mass m is dropped from rest from a height h above the ground. What is the instantaneous power generated by the gravitational force on the ball as a function of time? [a] The power is constant because the gravitational force is constant. [b] The power is zero because the gravitational force is a conservative force. [c] P(t) = mgh/t [d] P(t) = mg 2 t [e] P(t) = mg 2 t 2 /2 [1E.] A child pulls a sled of mass m across level ground at a constant velocity with a massless rope that makes an angle θ above the horizontal. The tension in the rope is T. Assuming the acceleration of gravity is g, what is the coefficient of kinetic friction between the sled and the ground? [a] (T/mg)tanθ [b] (T/mg)cot θ [c] (T/mg)sin θ [d] T cos θ/(mg T sin θ) [e] (T/mg)cos θ

5 [2.] The potential energy of a mass m moving along the x-axis is given by U(x) = (2 J)cos(2πx), where x is measured in meters and m = 3 kg. [a] Determine the force on the mass as a function of x, i.e. F(x). [b] What is the work done by the force in moving the mass from x = 0 m to x = 0.5 m. [c] For part [b], assume the mass is given an initial velocity v 0 = 4 m/s. What is the velocity when it reaches x = 0.5 m? [d] Give the position of at least one stable equilibrium point and one unstable equilibrium point. (Hint: it is useful to plot U(x).)

7 [3.] A block of mass m resides on a frictionless surface and is attached to two identical springs, each with spring constant k. When both springs are in their relaxed state, the mass resides at the origin, x = 0. In the initial state shown below, the mass is moved to position x 0 and released with no initial velocity, v 0 = 0. Your answers should be given in terms of k, m, x 0, and g. [a] What is the acceleration of the mass as soon as the mass is released? [b] What is the maximum speed that the block obtains? [c] Reconsider the problem assuming that there is a finite coefficient of kinetic friction µ k between the mass and the surface. What does µ k have to equal such that the mass only moves from its initial postion, x 0, back to the origin and stops. (Hint: Use energy considerations.)

9 [4.] Two point-like masses each of mass m are attached to each other by a massless rope of length L. The mass closest to the origin is attached to the origin by another massless rope of length L. The masses are rotated around the origin on a horizontal, frictionless plane at a constant angular speed ω in the counter-clockwise direction. The following picture is a view looking down on the tabletop from above. Your answers should be given in terms of m, L, and ω. [a] What is the tension T 2? [b] What is the tension T 1? [c] If the mass closer to the origin has a kinetic energy K 1 and the mass further from the origin has a kinetic energy K 2, what is the ratio K 1 /K 2?

11 [5.] A mass m starts from rest at a height R above the ground and travels down a frictionless, circular arc of radius R. At the bottom of the arc the mass heads up an inclined plane that makes an angle α with the horizontal. The coefficients of kinetic friction and static friction between the mass and the inclined plane are µ k and µ s, respectively. Your answers should be given in terms of g, R, θ, α, and µ k. [a] What is the speed of the mass at any point along the arc in terms of θ, R and g? [b] How far up the ramp, L, will the mass travel before stopping? [c] Once the mass stops, what is the minimum value of µ s such that the mass doesn t slide backwards?

Physics 1 Second Midterm Exam (AM) 2/25/2010

Physics 1 Second Midterm Exam (AM) 2/25/2010 Physics Second Midterm Eam (AM) /5/00. (This problem is worth 40 points.) A roller coaster car of m travels around a vertical loop of radius R. There is no friction and no air resistance. At the top of