TOPIC B: MOMENTUM EXAMPLES SPRING 2019 (Take g = 9.81 m s 2 ). Force-Momentum Q1. (Meriam and Kraige) Calculate the vertical acceleration of the 50 cylinder for each of the two cases illustrated. Neglect friction and the masses of the ropes and pulleys. 50 80 50 80 x 9.81 N Q2. The two blocks shown start from rest. The horizontal plane and the pulley are frictionless and the mass of the pulley and cords is negligible. Determine the acceleration of each block and the tension in the cord. 100 300 30 N F Q3. A weight 30 N hangs in the loop of a light inelastic cable, one end of which is pulled horizontally with force F = 15 + 3 e t N at time t s. If the weight starts from rest at t = 0, find: its speed after 2 s; the time taken for the weight to rise 2 m. Q4. (Exam 2016) A mass m = 2 hangs in the loop of a light, smooth inelastic cable, one end of which is pulled horizontally with time-varying force F = 19.62t, where F is in newtons and t is in seconds. The mass starts from rest at t = 0. F Show that the mass initially moves downward and determine for what period of time it continues to do so. Find the time at which the mass returns to its starting position, and its speed and acceleration at this time. m mg Mechanics Examples for Topic B (Momentum) - 1 David Apsley
Friction Q5. Determine the tension T in the cable which will give the 50 block a steady acceleration of 2 m s 2 up the incline in each case below. (Neglect the mass of cables and pulleys and assume the pulley is smooth.) T T 30 o 30 o 50 = 0.25 30 o 50 30 o = 0.25 Q6. (Meriam and Kraige) Compute the acceleration of the block for the instant depicted. Neglect the masses of the ropes and pulleys. = 0.4 40 30 o T = 100 N Q7. A gravity dam of mass 3 10 5 per metre length and with a vertical front face holds back water of depth 12 m. Assuming that the only vertical force is the weight of the dam, what minimum coefficient of friction is necessary to stop the dam sliding under the hydrostatic pressure of the water it is holding back? Why is this likely to be an underestimate for the required coefficient of friction? 12 m 300 000 /m Impulse-Momentum Q8. A car with a mass of 1800 is driven down a 5º incline at a speed of 90 km h 1 when the brakes are applied, causing a constant total breaking force of 6000 N. Find the time required for the car to come to a stop. Use the impulse-momentum principle. Mechanics Examples for Topic B (Momentum) - 2 David Apsley
Collisions Q9. (Exam 2013) Two vehicles are approaching a road junction, both moving with speed 45 kph (12.5 m s 1 ). One vehicle has mass 1000 and the other mass 750, and the roads meet at an angle of 60º as shown. The vehicles collide and initially move as a single body. (c) Calculate the velocity and momentum of the individual vehicles before the collision, using the x-y coordinate system shown. Calculate the velocity of the two vehicles immediately after the collision (treating them as a single body). After the collision, the total frictional force on the two vehicles is 6000 N, in the opposite direction to their combined velocity. How far do the vehicles travel after the collision? 1000? y 12.5 m/s 60 o x 750 12.5 m/s Q10. A squash ball is hit horizontally at 20 m s 1 from a height of 1.5 m directly toward a vertical wall 6 m away. Neglecting air resistance, and assuming a coefficient of restitution e = 0.7 between ball and wall, where, and at what speed, does the ball first hit the ground? Q11. A footballer kicks a ball at an angle of 35 to the ground towards a vertical wall 25 m away. Air resistance may be neglected. At what speed does he kick the ball if it hits the wall 10 m above the ground? If the coefficient of restitution between ball and wall is 0.6, how far back from the wall does the ball first bounce on the ground? Q12. A 10 package drops from a chute into a 24 cart with a speed of 3 m s 1 at an angle 30º below the horizontal. If the cart is initially at rest and can roll freely, determine: the final velocity of the cart; the impulse exerted by the cart on the package; (c) the fraction of the initial energy lost in the impact. 30 o 3 m/s Mechanics Examples for Topic B (Momentum) - 3 David Apsley
Q13. A ball is thrown at an angle of 70º from a height of 1 m above floor level (point A in the figure). The ball follows the trajectory shown and reaches a maximum height at B, before impacting the frictionless wall at point C. It then y rebounds in a direction at tan 1 (1/3) to 1 m x the wall. Model the ball as a point particle and assume that air resistance is negligible. Compute the initial speed required to reach point C. (c) Find the velocity vector just before hitting C in the x-y coordinate system shown. By considering the direction of the rebound, find the coefficient of restitution for the impact between the wall and the ball. A 70 o 6 m B tan = 1/3 C 3 m ram pile 2 m drop 0.1 m rebound Q14. The ram of a pile driver has mass 800 and is released from rest at a height of 2 m above the top of a 2400 pile. If the ram rebounds to a height of 0.1 m after a direct central impact with the pile, determine the following: the velocity of the pile immediately after impact; the coefficient of restitution; (c) the percentage of the energy lost in the impact. Q15. (Meriam and Kraige) n spheres of equal mass m are suspended in a line by wires of equal length so that the spheres are almost touching each other. If sphere 1 is swung aside and released and hits sphere 2 with speed v 1, find an expression for the velocity v r of the r th sphere immediately after being struck by the one adjacent to it. The common coefficient of restitution is e. 1 2 3 4 n v 1 2 1 u 1 Q16. Sphere 1 has velocity u 1 = 6 m s 1 in the direction shown and collides obliquely with sphere 2 of equal mass and diameter, which is initially at rest. If the coefficient of restitution is e = 0.6, find the speed and direction of each sphere following impact, and the percentage loss of energy in the collision. Mechanics Examples for Topic B (Momentum) - 4 David Apsley
Q17. (Exam 2014; extended by (d)) When struck, a golf ball of mass 0.045 is given an impulse of 1.6 N s. Find its initial speed. The golf ball travels a horizontal distance of 120 m over flat ground before bouncing. Ignoring air resistance, find the two possible angles to the ground with which it could have been struck. (c) The ball strikes a wall 65 m beyond the first bounce. Assuming the smaller of the two angles in part, and a coefficient of restitution e = 0.6 between ball and ground, at what height does the ball hit the wall? (d) Sketch graphs of the horizontal and vertical velocity components as functions of time, up to the point where the ball hits the wall. Indicate important values. wall 120 m 65 m Q18. (Exam 2016) Particle A, of mass 1.2, is attached to point O by a light, inelastic string of length 2.5 m, which is initially taut and horizontal. When released, particle A swings down and hits particle B, which is at rest on a small, elevated platform. The coefficient of restitution between particles is 0.6. (c) (d) 1.2 Find the speed of particle A immediately before the collision. 1.6 m State the magnitude and direction of the acceleration of particle A: (i) immediately after release; (ii) immediately before collision. If particle A is at rest after the collision, find the speed of particle B immediately after the collision and the mass of particle B. Determine how far particle B travels horizontally before hitting the ground 1.6 m below. A 2.5 m O 2.5 m B Mechanics Examples for Topic B (Momentum) - 5 David Apsley
Centre of Mass 0.5 m 0.1 m Q19. Find the centroid of the asymmetric rectangular frame shown. 0.05 m 0.25 m 0.7 m 0.25 m 5 cm o o 45 30 Q20. Find the centroid of the plane figure shown. 16 cm Q21. The uniform triangular lamina ABC shown has weight 30 N and is right-angled at B; AB = 300 mm and BC = 150 mm. The lamina is suspended by vertical light strings PA and QB and hangs in equilibrium in a vertical plane with AB horizontal and BC vertical. Find the tensions in the strings. The string QB is now cut and the lamina settles in equilibrium supported only by the string PA. What angle does AB make with the vertical? P A Q B C O 0.7 m 0.3 m L Q22. A uniform rectangular lamina has long side L and short side 0.7 m and has a circular hole of radius 0.3 m cut from it. The hole is equidistant from each of three sides as shown. When hung freely from a corner near the hole the long sides make an angle of 20 with the vertical. Calculate the length L. Mechanics Examples for Topic B (Momentum) - 6 David Apsley
0.05 m 15 o Q23. A uniform solid cylinder of radius 0.05 m is held on a rough plane inclined at 15 to the horizontal. The coefficient of friction between plane and cylinder is 0.3. If the cylinder is then released what happens if the height of the cylinder is: 0.4 m; 0.35 m. Q24. (Exam 2015) A stone archway consists of two uniform rectangular columns, surmounted by a semi-circular arch. The dimensions of the front of the arch are given in the figure and it extends a uniform and unspecified depth D behind. The density of the stone is 2600 m 3. Find the average pressure that the archway exerts on its foundations. Find the height of the centre of mass. 0.8 m 1.6 m 3 m Data: the centre of mass of a uniform semi-circular lamina, radius R is a distance above its base. 0.8 m 0.8 m Q25. A deep container has a square section with side 0.4 m. It is placed on a surface sloping at 25º to the horizontal, with two sides parallel to the line of greatest slope. The container is then slowly filled with water. The mass of the container itself and the thickness of its walls may be neglected. What minimum coefficient of friction is necessary between container and surface to prevent sliding? Assuming that the container does not slide, what volume of water must be poured into it before it topples over? 25 o 0.4 m Mechanics Examples for Topic B (Momentum) - 7 David Apsley