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Chapter 2 Dynamics Kinematics, the study of how things move, can only take a person so far. It was up to Isaac Newton, who was born the year Galileo died, to develop what has come to be known as “classical dynamics” which is the study of why things move as they do. He collected three basic laws of dynamics. 1. Objects in uniform motion will stay in uniform motion forever, unless acted upon by a net outside force. Uniform motion means ___________________ with a ____________________ Net force means _________________________ Outside forces are the only ones that wan cause an object to change its motion. 2. Force is the cause of all change in motion Fnet = m a F is force measured in ________________ (or ___________ in English units) It is the amount of _______ or _________ being exerted on an object. m is mass measured in ________________ (or ___________ in English units) It is the measure of the amount of ______________________. 3. For every action there is an equal but opposite reaction This is the most difficult of the three laws to really understand so We will put off studying this one in detail until we look at momentum 1 Review of projectile motion Let’s look at the example of a cannonball fired at certain angle above the horizontal. According to Newton’s first law of motion and in the absence of gravity, the cannonball should travel in a straight line forever. However, we live on Earth and there is gravity. It acts straight downwards. Each second the cannonball remains in the air it is pulled further and further away from it’s original path. This distance can be calculated by using “Dickie” d = ______ + ______ + ___ where “a” is _____ m/s2. If we neglect wind resistance the shape of the cannonball’s path is predictable. It follows along a parabolic path. 2 Mechanical Forces Notice: Force is a vector. It has both magnitude and direction Force (Newtons) has sub-units (kilograms x meters / second2) Gravity Fg The force of gravity acting on an object. (your weight) Fg Fslope =mg The portion of the force of gravity that is directed along a slope. (sliding force) Fslope = m g sin Fnormal Fg directed perpendicular to the surface. (Your apparent weight on the slope) Fnormal = m g cos Friction Ff = Fnormal Fno r q F g Fslope q is the coefficient of friction a measure of the stickiness between two surfaces. It ranges between zero (no friction) and 1.0 for most objects but may be higher. It is around 30,000 where two metal blocks have been welded together. Most ‘s are less than 1 1. Friction always works in opposition to the net outside force. 2. There are two types of friction Static and Kinetic. where the materials are sliding over each other and there is no sliding. 3. Important: s > ke 3 kinetic applies static where Lets Look at a Simple Elevator Problem The elevator is resting at the bottom of a building (A). The motor turns on to give it increasing upward speed (B). The elevator moves upward with constant speed (C). The motor works in reverse to slow the elevator down (D) The stopped elevator rests to let the passengers out (E). Lets fill in the graphs below. 4 A) A 100 kg block is being held up by a rope. D) A 100 kg elevator is moving upward with a speed of 5 m/sec B) A 100 kg elevator is accelerating downward at 5 m/sec2 (0.5 g’s) E) Wheezo the 100 kg cat is sliding down a rope which will brake if he applies more than 445 N force (about 100 lb) tension to it. What is the minimum acceleration he can have? C) A 100 kg elevator is accelerating upward at 5 m/sec2 (0.5 g’s) F) In the problem below a small 1000 kg car has brakes which can supply a braking force of about 445 N (about 100 lb). If the car is initially going 30 m/sec. What is its acc and how far will it slide before coming to a rest? 5 Pulley Problems Pulley problems are just like the train problems only now each object applies a force to the system because of gravity. Sometimes it moves the system forward (pos. force) and sometimes it works against the system (neg. force). In the following problems assume that pos. is in the direction the bigger object wants to fall. 1. Assume you have the two objects below find the acceleration of the system. 3. Now that you have found the acceleration of the system find the tension in the rope. pulley 20 kg 20 kg 30 kg 30 kg pulley a = _________ m/s2 2. . Frope = ___________ N Assume you have the two objects below find the acceleration of the system. 4. Now that you have found the acceleration of the system find the tension in the rope. a = _________ m/s2 Frope = ___________ N 6 Pulley Lab Comparing theory to reality Set up a pulley to your data acquisition equipment. You will be measuring the acceleration of systems when the masses on each side are not equal. Mass A Mass B Average Acceleration ( + or - ) 250 gr 200 gr _____________ ____________ 450 gr 400 gr _____________ ____________ ______gr _______gr _____________ ____________ ______gr _______ gr _____________ ____________ Now lets compute the theoretical acceleration that we would have predicted Mass A Mass B Theoretical Acceleration 250 gr 200 gr ____________ 450 gr 400 gr ____________ gr gr ____________ gr gr _____________ Explain your differences. 7 Multiple bodies In the pictures below complete the force diagram and calculate the force. (Assume plus is forward) 1. A model railroad train accelerates at a rate of 2.0 m/sec2 What net force must the engine exert? 60 kg 2. Find the tension in the first coupler if the train is accelerating at 2.0 m/sec2? 3. Find the tension in the second coupler if the train is accelerating at 2.0 m/sec2? 8 60 kg 100 kg 60 kg 60 kg 100 kg 60 kg 60 kg 100 kg You are pushing the blocks below over a frictionless surface. What force must you exert to maintain an acc. of 2.0 m/sec2? Fapp 100 kg 5. What is the force the 100 kg block exerts on the 20 kg block? 20 kg 60 kg 6. What is the force the 20 kg block exerts on the 60 kg block? 100 kg 20 kg Fapp 60 kg 100 kg 20 kg 4. 60 kg Fapp 9 Slope Problems In the pictures below complete the force diagram for a 100 kg block. and calculate the force. (Assume plus is forward down the hill) 1. Find the acceleration of the block and its Fapp = m g cos Fnet = m a Fslope = m a Fapp =ma a = ________ Fapp = ___________ Fno r q F g q Fslope As the slope becomes steeper the acceleration becomes greater and the object’s apparent weight becomes less. Calculate each of these for the following angles. Angle of slope =0 Acceleration Apparent weight = 10 = 20 = 30 = 40 = 50 = 60 = 70 = 80 = 90 10 From the information derived on the previous page, let us graph the relative accelerations and apparent weights as the steepness of the slope is changed. Calculating Friction To the right are three identical triangularly shaped blocks. Each is being pulled along the ground. Which do you think is hardest to drag? ___________________________ _____________________________________________________________________________ __ Scientists did a lot of studying of friction and determined that friction is directly related to two things. The force between objects (related to the object’s apparent weight) and the stickiness of the materials involved Ffriction = ( Fnormal ) general equation Ffriction = ( m g ) on flat ground Ffriction = ( m g cos ) on slopes _______________________________________________________________________________ 11 Problems with Friction Rubber tires on asphalt have a coefficient friction of 0.65 If you and your car (1200 kg) are moving along the road at 30 m/sec and you lock the brakes, how far will you slide? Picture below First find the acceleration of the car. Fnet = m a - FFrict = m a - m g = m a a = g = _______________________ Now knowing the acceleration we find the distance the car travels vf2 = vo2 + 2 a ( d ) 0 = 302 + 2 ______ ( d ) d = _______________________ 12 It turns out that there are two types of friction Kinetic (sliding) and Static (not sliding) Static friction is always greater than kinetic friction. This is why we have anti-lock braking systems, to prevent tires from sliding. In the case of the car above the static friction between tires and road is 0.99. Old fashioned locking brakes Initial velocity 10 m/sec 20 m/sec 30 m/sec 40 m/sec 50 m/sec Acceleration when sliding New antilock brakes Distance to stop Acceleration with antilock Distance to stop Graph both stopping curves. Do anti-lock brakes make a substantial difference? 13 Lets see what happens when you try to slow down a car while going down a 30 degree slope while having an initial speed of 30 m/sec. Assume that the coefficient of friction is 0.99 with our anti-lock brakes. First we must complete the vector diagram for the picture below. Givens: Fno r Fslope q F g q A. Find the acceleration of the car. B. Find the distance the car travels as it comes to a stop. 14 Putting on the brakes while going up a hill is a different story. Assume we are traveling up a 30 degree hill with an initial speed of 30 m/sec. Given a coefficient of friction of 0.99 we want to find the distance it takes to come to a stop. Givens: Fno r Fslope q F g q C. Find the acceleration of the car. D. Find the distance the car travels as it comes to a stop. Review: Stopping distance for a car with initial velocity of 30 m/sec Up hill __________________ 15 On the flat __________________ Down hill __________________ 1. An object is stationary on a horiz. surface ( = 0.9) A force is applied as shown. Complete the freebody diagram then calculate the force of friction. 2. An object is sliding on a horiz. surface with a speed of 5 m/sec when you first see it. There is friction ( = 0.9) Complete the freebody diagram then calculate the acceleration of the block. 3. An object is sliding up an inclined surface with a speed of 5 m/sec when you first see it. Complete the freebody diagram then calculate the acceleration of the block. There is no friction. The angle is 30 degrees. Find acceleration of the block. 16 4. An object is sliding up an inclined surface with a speed of 5 m/sec when you first see it. There is friction ( = 0.9) Complete the freebody diagram then calculate the acceleration of the block. The angle is 30 degrees. 5. An object is sliding down an inclined surface with a speed of 5 m/sec when you first see it. There is friction ( = 0.9) Complete the freebody diagram then calculate the acceleration of the block. The angle is 30 degrees. 6. An object is stationary on an inclined surface. There is friction ( = 0.9) Complete the freebody diagram then calculate the force of friction on the block. The angle is 30 degrees. 17 7. In the pulley system below find the acceleration of the blocks. Show a complete free body diagram for doing so) 8. In the pulley system below find the acceleration of the blocks. Show a complete free body diagram for doing so) 9. In the pulley system below find the acceleration of the blocks. The smaller block experiences friction ( = 0.9) Show a complete free body diagram for doing so) 18 How the Coefficient of Friction Is Determinied To determine the coefficient of friction, we drag known weight (mg) along the floor at a constant measure the force of pull. It will be equal in to the force of friction. We then substitute into g ) a mass of speed and magnitude Ffrict = ( m linoleum floor. If you need to pull with a force of 44.5 Newtons (10 lbs) what is the coefficient of friction? 19 Putting it together 1. Our friend Fred the cat is car moving at 30 m/s. He sees a beautiful feline 40 meters (about 120 feet) away who is hitching a ride. He slams on the brakes ( = 0.6) Will he stop before he reaches her or will he run her over? ____________________________ 2. In the problem above. How many g’s is Fred pulling as he comes to a stop? ____________________________ 3. Two things limit a car’s movement up a hill. One is the power of the engine and the other is the friction between the tires and the road. Let us assume that our car has a very powerful motor. So the limiting factor will be friction with the ground. What is the steepest hill (degrees) that a car can drive up at a constant speed of 30 m/sec. if = 0.9 ____________________________ 4. A 10 kg cat is suspended from the ceiling by a thin rope. If the cat is stationary, what is the tension in the rope? ____________________________ 5. Does he have acceleration? ____________________________ 6. Is the acc 0, positive, or negative? ____________________________ 7. Is gravity acting on him? ____________________________ 8. The moment the rope breaks (bummer), does he have an initial velocity? ____________________________ 9. Does he have an initial acceleration? ___________________________ 10. How much time elapses until he hits the tank of piranha fish which is 3 meters below? ____________________________ 20 Pulleys with masses on slopes and with friction Given an 80 kg block on a 40 degree slope (with respect to the horizontal) which has a coefficient of 0.4. the block is attached to a pulley at the top of the slope. On the other side of the pulley hangs vertically a 200 kg block . A. Draw a complete picture of the situation. B. Find the direction of the net force on the 80 kg block if friction is not included. C. Which way will the force of friction be acting? ___________________ 21 D. Include friction and calculate the acceleration of the system. E. Find the tension in the rope. Include a new picture. Quick review questions 1. A zoo keeper devises a rubber-band gun to shoot food to a monkey who is too shy to come down from the trees. If the monkey does not move, should the keeper aim above, at, or below the monkey? A. ABOVE THE MONKEY B. AT THE MONKEY C. BELOW THE MONKEY 2. If in the problem above the monkey lets go the instant the banana is fired, if he wants to hit the monkey he should aim A. ABOVE THE MONKEY B. AT THE MONKEY C. BELOW THE MONKEY 22 3. You have two blocks each with a weight of 100 Newtons. (hint: not mass but weight) In the first situation you hang a 100 N block over the pulley, in the second situation you just pull with a force of 100 Newtons. (neglect frict.) A. The acceleration in the second case is greater B. The acceleration in the first case is greater C. The acceleration is the same in both cases 4. In deep space (no gravity) a pellet is fired into a spiral tube. When it emerges, what direction will it travel? 5. Two smooth balls of exactly the same size, one made of wood and the other of iron, are dropped from a high building to the ground below. Because the iron ball has a greater inertia its acceleration will be (neglect wind resistance) A. B. C. D. greater than that of the wooden ball less than that of the wooden ball the same as that of the wooden ball there is not enough information to determine this 23 6. If blocks A and B each have a weight 100 Newtons and they are initially at rest, the tension in the rope is A. B. C. D. E. Zero Newtons 0 N. < tension < 100 N. 100 N. 100 N. < tension < 200 N. 200 N. 7. In the problem above, everything is the same but the blocks are initially moving with a speed of 2 m/sec. Now the tension in the rope is A. B. C. D. E. Zero Newtons 0 N. < tension < 100 N. 100 N. 100 N. < tension < 200 N. 200 N 8. What if the blocks above are not of equal weight but block A is 50 Newtons and block B is 100 Newtons. Assuming they start from rest and there is no friction. What is the tension in the rope? A. B. C. D. E. Zero Newtons 0 N. < tension < 100 N. 100 N. 100 N. < tension < 200 N. 200 N 24