15.1 Objective 1. Calculate the linear momentum of a particle and linear impulse of a force. 2. Apply the principle of linear impulse and momentum. Applications A good example of impulse is the action of hitting a ball with a bat. The impulse is the average force exerted by the bat multiplied by the time the bat and ball are in contact. Is the impulse a vector? Is the impulse pointing in the same direction as the force being applied? Given the situation of hitting a ball, how can we predict the resultant motion of the ball? Continue… When a stake is struck by a sledgehammer, a large impulse force is delivered to the stake and drives it into the ground. If we know the initial speed of the sledgehammer and the duration of impact, how can we determine the magnitude of the impulsive force delivered to the stake? A hard lesson Another one Linear impulse Principle of linear impulse and momentum Example • A golf ball having a mass of 40 g is struck with a force profile F = 200 sin(100t). Find a) the ball’s velocity leaving the tee; b) the distance the ball will travel (assume flat range) https://www.youtube.com/watch?v=YqgyHQyll38 Example • Force on a 2g bullet, as it travels horizontally through the barrel, varies as shown. Neglect friction. Find: F0 (maximum net force) if the bullet velocity is 500m/s at t=0.75 ms. Example • A car traveling at 4 ft/sec (2.72 mph) weighing 2700 lb crashes into a wall. The duration of the impact is 0.06s. • Find: a) average impulsive force during collision if brakes are not applied • B) average impulsive force if all tires brake (k=0.3) https://www.youtube.com/watch?v=joMK1WZjP7g 15.2-15.3 objective 1. Apply the principle of linear impulse and momentum to a system of particles. 2. Understand the conditions for conservation of momentum. Applications This large crane-mounted hammer is used to drive piles into the ground. Conservation of momentum can be used to find the velocity of the pile just after impact, assuming the hammer does not rebound off the pile. If the hammer rebounds, does the pile velocity change from the case when the hammer doesn’t rebound ? Why ? In the impulse-momentum analysis, do we have to consider the impulses of the weights of the hammer and pile and the resistance force ? Why or why not ? Example • A train has one engine (50 Mg) and 3 cars (30 Mg each). It takes 80 seconds for the train to uniformly increase speed to 40 km/hr, starting from rest. • Find: a) the coupling force between the engine and first car, • B) the traction force of the engine (Assume all cars roll freely) https://lancastercatholichs.instructure.com/courses/410/files/16 64/download Non-impulsive forces and impulsive forces example • A barge B weighs 30,000 lb and supports a 3,000 lb car C (the barge is not secured to the pier). The car is driven 200 ft across the barge. Neglect water resistance. • Find: how far the barge moves from the pier Example • A 5 kg spring-loaded gun rests on a smooth surface. It fires a 1 kg ball with a velocity of 6m/s relative to the gun as shown. • Find: the separation distance d between ball and gun. Example • A barge weighs 45,000 lb and supports two cars A and B, weighing 4,000 lb and 3,000 lb, respectively. They start from rest and accelerate towards each other (aA = 4ft/s2, aB = 8 ft/s2) until they reach constant speeds of 6 ft/s (relatively to the barge). Initially, the barge is at rest. Neglect water resistance. • Find: the speed of the barge just before impact 15.4 Objective 1. Understand and analyze the mechanics of impact. 2. Analyze the motion of bodies undergoing a collision, in both central and oblique cases of impact. Collision Applications The quality of a tennis ball is measured by the height of its bounce. This can be quantified by the coefficient of restitution of the ball. If the height from which the ball is dropped and the height of its resulting bounce are known, how can we determine the coefficient of restitution of the ball? Continue… In the game of billiards, it is important to be able to predict the trajectory and speed of a ball after it is struck by another ball. If we know the velocity of ball A before the impact, how can we determine the magnitude and direction of the velocity of ball B after the impact? What parameters would we need to know to do this? Central and oblique impact Phases of central impact https://www.youtube.com/watch?v=QFlEIybC7rU Example • Disk A (2kg) slides on a smooth surface (VA1 = 5m/s) and strikes Disk B (4kg, VB1=2m/s) with central impact (e=0.4). • Find: VA2 and VB2 Example • A 2kg ball strikes a suspended 20 kg block with a velocity of 4m/s. The coefficient of restitution is 0.8. • Find: the height h to which the block will swing before it momentarily stops. Oblique impact Example • A girl throws a ball with horizontal velocity v1=8ft/s. The “e” between the ball and the ground is 0.8. • Find: a) the velocity after the ball rebounds, and • B) the maximum height the ball rises after the first bounce. 15.5-15.7 Objective 1. Determine the angular momentum of a particle and apply the principle of angular impulse & momentum. 2. Use conservation of angular momentum to solve problems. Applications Planets and most satellites move in elliptical orbits. This motion is caused by gravitational attraction forces. Since these forces act in pairs, the sum of the moments of the forces acting on the system will be zero. This means that angular momentum is conserved. If the angular momentum is constant, does it mean the linear momentum is also constant? Why or why not? Applications The passengers on the amusement-park ride experience conservation of angular momentum about the axis of rotation (the z-axis). As shown on the free body diagram, the line of action of the normal force, N, passes through the z-axis and the weight’s line of action is parallel to it. Therefore, the sum of moments of these two forces about the z-axis is zero. If the passenger moves away from the z-axis, will his speed increase or decrease? Why? Angular momentum Direction Example • Determine 0 for the 1.5 kg particle below: Newton’s 2nd law Example • Two spheres, each 3 kg, are attached to a rod of negligible mass. A torque M = 6e0.2t (N/m) is applied as shown, starting from rest. • Find: speed of spheres after 2 seconds Conservation of angular momentum An example of this condition occurs when a particle is subjected only to a central force. In the figure, the force F is always directed toward point O. Thus, the angular impulse of F about O is always zero, and angular momentum of the particle about O is conserved. Example • An amusement park ride consists of a car attached by cable to point O. It rotates in a horizontal plane, v1 = 4ft/s where r = 12 ft. The cable is then retracted at a constant rate of 0.5 ft/s. • Find: the speed of the car after 3 seconds. Example • A 0.1 kg block is given a horizontal velocity v1 = 0.4 m/s when r1 = 500 mm. It slides along a smooth conical surface. • Find: the block’s speed and angle when h = 100mm.
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