Lesson 33: Horizontal & Vertical Circular Problems

Transcription

1 Lesson 33: Hoizontal & Vetical Cicula Poblems Thee ae a wide vaiety of questions that you do if you apply you knowledge of cicula motion coectly. The tough pat is figuing out how to set them up. You need to figue out two things at the stat 1. Is the cicle the object is moving in hoizontal o vetical? 2. What combination of foces ae causing the centipetal foce? Hoizontal Cicle Poblems A hoizontal cicle is any cicle that lies flat paallel to the gound. A pefect example is a ca going aound a egula taffic cicle. This path can be a complete cicle o even just a potion of a cicle (as long as you can still measue basics like what the adius would have been if it had been a complete cicle). If the object is on the gound, it is vey common to say that the needed centipetal foce is caused by the foce of fiction with the gound. Example 1: A constuction team is looking at building an exit amp on the Anthony Henday feeway. They ae concened that cas should be able to go though the tun without skidding off the oad even if conditions ae bad. Detemine the minimum adius that can be used fo the cuve of the tun if a ca taveling 110 km/h might be taveling in conditions that cause a coefficient of static fiction of only 0.60 between the ties and the oad. Since the ca is moving in a cicle thee is a centipetal foce, caused by the fiction between the ties and gound. As the ties push outwads on the gound, the gound pushes inwads on the ties. You can see the foce of the ties pushing the gound outwads in situations when a ca is going aound a tun on a gavel oad; you can see the gavel being shot outwads fom the cicle. =F f =μf N We stat by substituting fomulas fo each side of the elationship. =μ F g Since we ae on a flat, level suface is equal to. =μmg This allows us to substitute in = mg. v 2 =μ g We can cancel both the m on eithe side, since they it is common to both tems and is found on top on both. = v2 μg = (9.81) = =1.6e2m This allows us to manipulate the fomula fo, substitute in ou values, and solve it. This is a huge tun the diamete would be ove 300 m! This would not be easonable to constuct. 11/16/2015 studyphysics.ca Page 1 of 5 / Section 5.2

2 Example 2: The easie thing to do than build a huge exit amp is to simply ask dives to slow down a bit. This is why exit amps always have a posted speed limit less than the feeway. The Anthony Henday feeway has exit amps with a diamete of about 180m. Use this numbe and a coefficient of static fiction of still 0.60 to detemine the maximum speed that is safe on an exit amp. Eveything fo this poblem is the same as Example 1 until you get to what you solve it fo, so I'll just skip the fist few lines. Don't foget we need to use adius, not diamete. v 2 =μ g v= μ g v = 90.0(0.60)(9.81) v=23.0m/s This is about 82.9 km/h, which is much moe easonable than the speed we used in Example 1. Vetical Cicle Poblems A vetical cicle is any cicle that is pependicula to the gound. Although they cetainly do not move fast, feis wheels ae a geat example of a vetical cicle. Many high speed olle coastes also make use of at least one vetical cicle. These also make a geat example fo physics questions because of the elationships between the foces involved. Vetical Cicle Consideations To successfully solve a vetical cicle poblem you will need to keep a few things in mind. Although this list seems huge, they ae all following the egula ules fo the diections and signs foces should have. Fo the object to move in a cicle, thee must be a centipetal foce acting on it. Examine the paticula situation you ae looking at and ty to decide which foce(s) ae aligned to add up to give you this special cente seeking vesion of net foce. Illustation 1: The London Eye, also known as the Millennium Wheel, is essentially a gigantic feis wheel. If the object is at the top, the centipetal foce (pointing like it always does, towads the cente) is down and negative. If the object is at the bottom, the centipetal foce is pointing up and positive. Foce due to gavity will often be something you have to conside. It is impotant to emembe that the foce due to gavity always points staight down. In a situation such as a olle coaste, you might be able to show that thee is a nomal foce exeted by the suface (like olle coaste tacks) the object is touching. If the object is nea the top, the nomal foce points down. If the object is nea the bottom the nomal foce points up. If the object is not on a suface, it might be spinning on the end of a sting o something like that. If this is the case, you will need to take into account the foce due to tension exeted by the sting on the object. 11/16/2015 studyphysics.ca Page 2 of 5 / Section 5.2

3 If the object is nea the top, the foce due to tension points down. If the object is nea the bottom the foce due to tension points up. top Foce Top of Cicle Bottom of Cicle Centipetal down up Gavity down down Nomal down up Tension down up o o bottom Example 3: A olle coaste is going though a loop that has a adius of 4.80 m. The olle coaste cas have a speed of 13.8 m/s at the top of the loop. Duing testing and development of the olle coaste, it was detemined that the cas and passenges have a combined mass of 4800 kg on an aveage un. Detemine the amount of foce the tack must be designed to withstand at the top in ode to keep the cas going aound the loop. Thee ae two foces that will be acting on the cas at the top of the loop. The foce due to gavity will be pulling it down (towads the cente). The nomal foce of the tacks pushing against the cas will be pushing the cas down (towads the cente). These ae the two foces that combined will exet the necessay net foce, the centipetal foce, to keep the olle coaste moving in a cicle. Notice that the centipetal foce is also pointing downwads, which we will have to take into account when we substitute into the fomula... Illustation 2: Diection of foces at top and bottom. Illustation 3: The ca going though the top of the loop. We can say that the necessay stength of the olle coaste tack is eally the nomal foce, since that is the amount of foce that the tack must exet against the ca (just like a table must be stong enough to exet a nomal foce up against a heavy box placed on the table). 11/16/2015 studyphysics.ca Page 3 of 5 / Section 5.2

4 Notice the negative sign we put in font of the numbes substituted in fo the centipetal foce, since we know the centipetal foce needs to be negative (pointing down). + = = mv2 mg = 4800(13.8)2 4800( 9.81) 4.80 = = 1.43e5 N The tack must be able to exet a foce of 1.43e5 N [down] at the top of the loop. If it is not stong enough to exet a foce this lage, the tack will beak. Example 4: The tack fo the olle coaste mentioned in the last example needs to actually be stonge at the bottom of the loop. Although the cas actually speed up as they come down to the bottom of the loop, assume the same velocity, adius, and mass as Example 3 and detemine the amount of foce the tack must be able to withstand at the bottom of the loop. We'e going to calculate nomal foce again fo the stength of the tack, and needs to be quite stong now, since it must suppot the ca against the foce of gavity and supply the centipetal foce needed to keep it moving in a cicle. The magnitudes fo and stay the same. Illustation 4: The ca at the bottom of the loop. We now see and point up (positive), but still points down (negative). + = = mv2 mg = 4800(13.8)2 4800( 9.81) 4.80 =237528=2.38e5N Example 5: Detemine the minimum speed the cas on this olle coaste can move in ode to just baely make it though the loop at the top. Going at the minimum speed means that the cas will just baely make it though the top. The foce due to gavity will supply all the centipetal foce needed to keep moving though the cicle. Thee doesn't have to be any nomal foce supplied by the tack at all; with nomal foce equal to zeo we can just cancel it so we'e left with only and. 11/16/2015 studyphysics.ca Page 4 of 5 / Section 5.2

5 Also, since we'e at the top, and ae both pointing down, so they'e both negative. We'll just ignoe the negatives on both sides, since they'll just cancel out anyways. =mg v= g v = 4.80(9.81) v=6.86m/s If you need to solve a poblem involving an object spinning on the end of the sting, you can solve it in a simila way to the questions above. The only thing you eally need to change is to use foce of tension in the sting instead of nomal foce. Example 6: A 245 g mass is on the end of a 35 cm long sting. Detemine the tension in the sting at the top if the mass is spinning at 5.67 m/s. + = = mv2 mg = 0.245(5.67) ( 9.81) 0.35 = = 24.9 N Notice thee is a minus sign inseted on the centipetal foce tem. This is because we know centipetal points downwads when we ae at the top of the cicle. Homewok p.259 #1-3 p.262 #1 p.264 #1 11/16/2015 studyphysics.ca Page 5 of 5 / Section 5.2