Faculty of Engineering and Department of Physics Engineering Physics 131 Midterm Examination Monday February 24, 2014; 7:00 pm 8:30 pm

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1 Faculty of Engineering and Department of Physics Engineering Physics 131 Midterm Examination Monday February 4, 014; 7:00 pm 8:30 pm 1. No notes or textbooks allowed.. Formula sheets are included (may be removed). 3. The exam has 9 problems and is out of 50 points. Attempt all parts of all problems. 4. Show all work in a neat and logical manner. Questions 1 to 4, 8(a) and 9 do not require detailed calculations and only the final answers to these questions will be marked. For the rest of the questions, details and procedures to solve these problems will be marked. 5. Write your solution directly on the pages with the questions. Indicate clearly if you use the backs of pages for material to be marked. 6. Non-programmable calculator allowed. Turn off all cell-phones, laptops, etc. and put them in your backpack. DO NOT separate the pages of the exam containing the problems. LAST NAME: FIRST NAME: ID#: Please circle the name of your instructor: B01: Kaminsky B0: Beach B03: McDonald B04: Tang B05: Wheelock B06: Ropchan 1

2 Please do not write in the table below. Question Value (Points) Mark Total 50

3 1. [5 Points] Cars A and B begin at distances s A = 1.0 km and s B = 1.5 km from an intersection, as measured bumper to stopping line. The vehicles are initially travelling at 18 km/h and 36 km/h, respectively, and they are accelerating at constant rates a A = 0.05 m/s and a B = 0.10 m/s. Neither vehicle, however, will exceed the posted speed limit of 54 km/h. Each lane is 4.0 m wide. Expressed in a coordinate system centered on the intersection, and whose axes are aligned with the two roads, the vehicle positions are r A = ( s A 4.0,.0) m and r B = (.0, s B +4.0) m. The distance between the vehicles is r A/B = r A r B. Of the 11 lines shown on the three graphs below, 5 correctly describe the evolution in time of s A, s B, v A (speed of A), v B (speed of B) and r A/B. Match those 5 lines to the corresponding quantity by writing the appropriate numbers in the blanks provided. s A is described by curve 3 s B is described by curve 4 v A is described by curve 7 v B is described by curve 5 r A/B is described by curve 9 3

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5 . [ Points] A baseball was thrown from a cliff and the projectile motion was simulated by using a computer. The relationship between the vertical position (in the y-direction) and the horizontal position (in the x-direction) is shown graphically below. One of the curves was generated for the case when there was no air resistance, and the other curve was generated when air resistance was present. The initial velocities and angles of the baseball are the same. Answer the following questions regarding the curves and the motion. (a) [1]Which curve represents the motion when air resistance was present? Circle your choice. Curve I Curve II (b) [1]The curves were drawn and extended below the line y = 0. What does this mean physically? Circle your choice. (A) The balls stopped moving. (B) The balls changed direction of motion. (C) The balls fell below the height from which they were thrown. (D) Insufficient information to answer. 5

6 3. [5 Points] The block is initially given a push, then it slides down a frictionless track (under the influence of gravity) and remains in contact with the track all the way around the loop. Five locations along the track (A, B, C, D, E) are indicated on the diagram. In the table below, circle the location(s) (A, B, C etc.) that have the following attributes. If there are no such locations, circle none. Tangential acceleration (a t ) = 0 at A B C D E none Normal acceleration (a n ) = 0 at A B C D E none Normal component of velocity (v n ) is nonzero at A B C D E none Speed is decreasing at A B C D E none Acceleration and velocity vectors are perpendicular at A B C D E none Tangential acceleration is zero when the speed is constant (point C), or when speed reaches a local minimum (point E) or maximum (N/A for the 5 locations) value. Normal acceleration is zero when the speed is zero (N/A for the 5 locations) or the radius of curvature is infinite (i.e., where the path is straight) as at point C. Velocity is always tangential to the path and there is no normal component. Speed is decreasing when the block is sliding upward (point D). Since velocity is only in tangential direction, it is perpendicular to acceleration when acceleration only has normal component (point E). Note at point C, acceleration = 0. 6

7 4. [ Points] In the figure shown below, the pulley is massless and frictionless, the rope is massless, and 45. Block B moves down the incline with a constant speed v B. The relative velocity of B with respect to A is given by v B/A and has components (v B/A ) x and (v B/A ) y with respect to the coordinate system shown. Now consider a similar arrangement with the angle of the incline replaced by and the masses of blocks A and B adjusted so that block B moves down the incline with the same speed v B. Using the same coordinate system, do the following quantities increase, decrease, remain the same, or is it impossible to say? Circle one answer for each. (v B/A ) x Increase Decrease Remain the same Impossible to say (v B/A ) y Increase Decrease Remain the same Impossible to say (v B/A ) x Increase Decrease Remain the same Impossible to say (v B/A ) y Increase Decrease Remain the same Impossible to say In both cases, the motion is constrained. With the datum in the centre of the pulley: s s l constant v v 0 v v. A B A B A B Thus, B moving down the incline means A moving to the right, and since v B doesn t change, neither does v A, so in both cases va vbi. Initially at angle : vb vbcosisinj and B A B A B Hence, ( v ) v cos,( v ) sin. B/ A x B B/ A y v / v v v cos i sin j. As : ( v ) v cos,( v ) sin. B/ A x B B/ A y Since 0 45, both ( vb/ A) x and ( v B/ A) y are negative in the two cases. In addition, cos cos and sin sin. Hence, both components become more negative as, and so both decrease. However, since they both decrease, they both increase in absolute value. 7

8 5. [7 Points] A body travels along a smooth horizontal surface in the positive x direction and is subject to a quadratic drag force whose magnitude also increases linearly with position. Thus, the body s acceleration is given by: a = xv m/s (with x in m, and v in m/s). At t = 0, the body is at x = 0, and its velocity v is +4 m/s. (a) [] In terms of M, L and T, what dimensions must the carry? (b) [5] Find the velocity of the body as a function of position. (Hint: you can still separate the appropriate variables as always.) (a) (b) 8

9 6. [7 Points] On a planet where g = 10 m/s, a football is kicked from the origin (point A) at a speed of 6.0 m/s, directed θ A = above the horizontal (so cos θ A = 3/5, sin θ A = 4/5). At t = 0.75 s, what is the football s: (a) rate of change of speed; (b) normal component of acceleration; (c) radius of curvature. Method 1: Method : 9

10 Method 3: Method 4: v 3.6 m/s, v t x y v v v t x y a t dv dt t t At t 0.75 s, at 6 m/s a g a m/s n t At t 0.75 s, v 3.6 m/s, v.7 m/s x y v / a / 8.53 m n 10

11 7. [8 Points] The car travels along a circular path having a radius of 30 m at a speed of v 0 = 5 m/s. At s = 0, it begins to accelerate with v = dv/dt = (0.05 s) m/s, where s is in meters. Determine its speed and the magnitude of its total acceleration when it has moved s = 18 m. a ds vdv 0.05sds vdv t s sds v v 0 vdv 0.05s v v 0 v s 0.05s v 0.05s 5 m/ s 0 v m/ s a t m/ s a n v m/ s 30 a a a 1.64 m/ s n t 11

12 8. [8 Points] Consider the system shown. Both blocks move at a constant speed. The relative velocity of B with respect to A, v B/A, is directed downwards towards the right at an angle as measured from the usual positive x axis. (a) [1] Does B move up or down the incline? Circle one answer. Up Down If the speed of B is 3 m/s determine: (b) [3] the velocity of A, expressed as a magnitude and direction with respect to the nearest horizontal; and (c) [4] the angle that v B/A makes with respect to the nearest horizontal. (b) Placing datum at the centre of the top pulley: s s l constantv v 0. A B A B v 3m/s down the incline 3m/s relative to the datum. B v B So v v 6m/s = 6 m/s towards the datum: v 6 m/s. A B A (c) v 6 m/s 6 j A v 3 m/s B 3 cos60 i sin 60 j v v v 3cos 60i 3sin 60j 6j1.5i8.598 j. BA B A Angle : tan

13 9. [6 Points] Two stacked blocks are traveling horizontally along a rough horizontal surface as shown when the bottom block encounters a spring anchored on one side to a wall. The surface between the two blocks is also rough. While the spring is being compressed and while the top block remains at rest relative to the bottom block (figure on the right), draw the complete free-body and kinetic diagrams for both blocks. Be sure to clearly indicate the correct directions of any vector on each diagram. All symbols shown in your diagrams are assumed to take on positive values. In addition, YOU MUST USE the following symbols for forces (IF APPLICABLE): m i g for gravitational forces N ij for normal forces i acting on j F spring for elastic restoring forces f k for kinetic frictional forces f s for static frictional forces T ij for tensional forces Free Body Diagram for A: Kinetic Diagram for A: Free Body Diagram for B: Kinetic Diagram for B: 13