Supplementary problems - Circular motion

Problem solving strategy
1. Draw a picture
2. Draw and label the forces acting on the object
3. Resolve all forces as necessary into components that are parallel and perpendicular to the line drawn from the object to the center of the circle (this is a radius).
4. Combine all forces parallel to a radius. The forces add if they point in the same direction; they subtract if they are oppositely directed.
5. Like all other problems the vector sum of the forces = F
net = ma which in this case ma = mv*2/R.

1. A civil engineer is in the process of designing a limited-access roadway from town A to town B. Because of natural obstructions in the landscape (perhaps a special stand of trees or some interesting land feature), the road will have a curve in it. The engineer takes into account the following factors: 1) the road way will be essentially flat; 2) the minimum coefficient of friction that one can guarantee is u = .25; 3) The radius of curvature for this road way can be no more than 100 m. a) what should be the maximum speed of any vehicle if it is to stay on the road? b) Assume that after this road way is built, a particularly bad ice storm reduces the coefficient of friction to u = .1. What lower speed should drivers undertake to ensure that they can negotiate the curve?

2. Tarzan (m = 100 kg) swings with help of a vine. Holding the vine, Tarz drops five meters, reaching the bottom of the arc of the swing with a speed of 14 m/s. What is the tension one the vine at the bottom of the swing? What fraction of his weight is this tension?

3. A car rounds a banked curve that is slick with ice. If u is reduced to u = 0, the speed cf the car is 60 mph (27 m per second), and the radius of the curve is 50 m. At what angle should curve the banked in order to guarantee that the car will negotiate the curve safely?

4. The child care operation down the hall has babies in small basket-like carriers. The handle makes it ever easier to grasp the unit and swing it into a vertical circle. Show that the minimum speed necessary at the top of the arc must be v' = [sqrt (Rg)] to ensure no accidents. Here R is the radius of the circle and g is the local acceleraltion due to gravity.

5. Longplaying records are played on a turntable rotated 33 1/3 RPM. These disks are about 25 cm in diameter. If we place a penny and the rim of such a record, what should be the coefficient of friction between the penny in the disk to ensure that the penny does not slip off?

6. Isaac Newton observed that the moon orbited the earth in a path that was nearly circular as opposed to a straight line. Because the path is not a straight line, the net force acting on the moon is not equal to zero. The moon underwent a centripetal acceleration such that a = v^2/R Let us calculate the value of the acceleration. A) Calculate the speed of the moon. The moon goes through one circumference in one month v = 2 pi R/T where R is the mean earth-moon separation (R = 3.82 x 10^8 m)and T is the orbital period of the moon (T = 2.36 x 10^6 s). B) Now calculate a = v^2/R. C) What's the big deal? Newton suggested that the same force that holds you to the ground also holds the moon in place. At one Earth radius, g = 9.8 m/s/s. At 60 earth radii away a = 9.8 m/s/s / 3600. Compare this solution to that of part 6a.

This page last edited 01/23//09


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