Solving Newton s Second Law Problems

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1 Solving ewton s Second Law Problems Michael Fowler, Phys 142E Lec 8 Feb 5, 2009 Zero Acceleration Problems: Forces Add to Zero he Law is F ma : the acceleration o a given body is given by the net orce on that body divided by the mass. But it s oten surprisingly diicult to ind the orce on a particular body. he strategy is to mentally isolate that body, and enumerate the orces acting on it this is the ree body diagram. hese vector orces must then be added correctly head to tail and the resulting total orce ound. his will give the direction o the body s acceleration. We ll begin with the static situation: no acceleration. For example, we can ind the tension in the ropes below by noticing that the total orce on the know where the ropes come together must be zero, since the know isn t moving. We draw a triangle o orces, the sides parallel to the corresponding ropes. he angles in this triangle are thereore determined by the angle o slope o the ropes, so with simple trig the ratio o to an be ound. otice that the orce vectors must all point the same way going around the triangle. Otherwise, they won t add to zero. his also works or side ropes at dierent angles, it s just more complicated. he knot where the three ropes meet is not moving, so the three tension orces acting on it must add as vectors to zero: see triangle o orces on right. he sides o the triangle are parallel to the corresponding ropes, so the tension can be calculated given the weight and the angles.

2 2 For a block at rest on a slope, the upward orce o the slope s surace on the block must exactly counterbalance the block s weight. his orce rom the slope has two components: the normal orce, meaning the orce perpendicular to the surace resulting rom the springiness o the surace, the weight has compressed it slightly and it s pushing back, and the rictional orce preventing the weight rom sliding down the slope. From the triangle o orces (all going round the same way, and thereore adding to zero!) one can ind, in terms o, since the angles o the triangle are determined by the slope. Block at Rest on Slope I the block remains at rest, the three orces acting orm a triangle as shown. he angle o slope is suicient to determine, given.

3 3 A steady velocity situation is o course the same as a static one: there s no acceleration, so the vector total o orces acting on the body must be zero. ake the case o a sled Pulling Sled at Steady Velocity I the sled is moving at constant velocity, the total orce on it is zero: the our orces add to zero as shown. being pulled using a rope at a ixed angle as in this diagram. Separating the orce rom the ground into normal and rictional components, we have a quadrilateral o orces. Since the sled is moving, (the coeicient o kinetic or sliding riction). his means that given the K angle o the rope, the tension needed to maintain speed can be ound. (Write out the equations or vertical and horizontal components.)

4 4 Object Accelerating: Forces Add to Mass Acceleration Consider now a block accelerating as it slides down a slope, with rictional resistance. he acceleration vector is directly down the slope, the rictional orce directly up the slope:. K Block Sliding down Slope ma I the block is sliding down, the three orces acting add up to ma, where a is the acceleration down the slope, as shown. he three orce vectors no longer add to zero, they add to ma. o solve this problem, equate the orce components parallel to the slope, then those perpendicular to the slope.

5 5 I a road going around a bend is suitably banked, the component o the normal orce pointing towards the center o the curve will contribute to the necessary v 2 /r acceleration. I the road is not banked, the centripital acceleration is generated entirely by sideways riction, and especially on wet roads this can be problematic. Car on Curved Banked Road at Design Speed mv 2 /r At the design speed, there will be no sideways riction. ote the normal orce is now greater in magnitude than the weight. However, or a given angle o banking, equating mv 2 /r to the horizontal component o the normal orce determines v: so it only works perectly or one speed, the design speed o that stretch o road, and at that speed you could make it round the curve on a sheet o ice. But or speeds anywhere near the design speed, less rictional orce is required on a banked road.

6 6 For a given coeicient o (static) riction, there is a maximum speed beore the car deviates rom the circular path. Here is the diagram or F ma : Car on Curved Banked Road at Maximum Speed mv 2 /r = μ S = μ S At the maximum speed, sideways riction = μ S. o solve the problem, ind and equate the horizontal and vertical components. Actually, i S 1 (realistic or some tires) and the road is banked at 45 degrees, the maximum sae speed, according to this analysis, increases without limit. (ry drawing the vectors!) However, we ve ignored the possibility that the car might begin to roll up the banking.

7 7 For a car accelerating uphill, the external orce in F ma causing the acceleration is the rictional orce acting uphill, minus the component along the slope o the weight. Car Accelerating up an Incline ma he rictional orce rom the road is pushing the car uphill. he total orce o the road on the car is o course +, the dotted green vector above. he dashed vector is the total orce rom the surace on the car, this vector plus gives ma. he orce rom the seat on the driver would be in exactly the direction o the dashed vector. Problem Solving Strategy ma It s all about solving F or a particular body. he irst thing is to igure out which body, and see what you can say about its acceleration: you might not know its magnitude, but you probably know its direction. I it s going in a circle at constant speed, you know it s accelerating towards the center. I it s going in a straight line, the acceleration must be along that line. (O course, there are trickier cases: a car picking up speed as it goes around a bend, a planet in an elliptical orbit, etc.) ow you enumerate the orces on the body, and use F F1 F2 F3 ma. By this I mean draw a diagram with the orces drawn as vectors, with the directions correct (you know normal orces are always perpendicular to the surace, riction along the surace) and try to represent this equation, taking or example three orces, as a head-to-tail vector sum equaling the ma vector, as in the examples shown above. Don t represent the vectors in terms o their components until you can see how they add up: this will give you a clearer picture o what s going on, without the clutter. Only then, to do actual calculations, put the vectors in terms o their components in two directions at right angles, and write, say, Fx max, etc.

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