Remember the framework for solving Newton's Law/Circular problems is available on the course web page under Web Resources. There is a need in many of these settings to understand the relationship between linear and angular speeds as well as the period and frequency associated with a circling/cycling object. Please refer to the breadcrumbs for reminders.

 

1) Determine the radius of the orbit and the speed of a geostationary satellite. Such a satellite remains located over the same geographic point on the earth as it rotates. You will need to look up Newton's Universal Law of Gravity, and consider its impact on your freebody diagram, and various other constants in your book. In Newton's Universal Law of Gravity the relevant radius is from the center of the earth. You may ignore all other gravitational forces:) You can get some sense of what your result will be from this image of space junk near the earth. The geostationary orbit is the crisp circular ring of debris.

2) Imagine you are sitting on a recently installed merry go round - MGR- (I hope you all know what this is) that is nice and level. This particular MGR is frictionless so that in order to stay on you have to tie yourself to the center with a string to keep from sliding off. To provide a sense of excitement you find a string which has a known breaking tension of 1000 N. The diameter of the MGR is 3 m and you are sitting on the edge. Take your mass to be 90 kg. How fast can I get going, both angularly (rad/s) and linearly (m/s), before the string breaks?

3) At one point the fastest CD-ROM drives spun at around 10,350 RPM for a 120 mm diameter CD. Consider the forces on a small cubical bit of the CD on the edge (this setting also applies to hard drives). When you sketch this cube you will find that 3 of its six sides are in contact with the rest of the CD. In your future engineering classes you will discuss shear and tensile forces within materials. Shear forces are like friction and act parallel to the imaginary surfaces you have drawn. Tensile forces act like ropes to hold the material together. Draw all the forces on each of the three faces of the cubical bit you have sketched. Two of the surfaces will have shear forces in two directions (vertically and horizontally) with no tensile forces. One of the surfaces will have a tensile force and a shear force. Which of these forces balance gravity and which are responsible of the net force in the radial direction that keeps the cube travelling in a circle? Assume the little bit of the CD has a mass of 1.2 mg. What is the tensile force required to keep it moving with the rest of the CD?

4) A conical pendulum is a ball on a string which is swinging in a horizontal circle. The angle between the string and the vertical direction is constant. Consider such a pendulum with a 20 kg ball on the end of a 3 m long string which makes an angle of 5o with the vertical as it swings. Find the tension in the string and the period of its motion.

5) You are needed to design a highway curve for traffic traveling at a constant 60 km/h. Due to constraints of the landscape the radius of the curve must be 150 m. What is the appropriate bank angle of the curve so there will be no problems on an icy (frictionless) day? If you design the curve as a flat curve what is the minimum coefficient of (static or kinetic?) friction needed?

6) As part of the same design team you are checking on the layout of a road which passes over the crest of a hill. In cross section the crest of the hill has a radius of curvature of 250 m. What is the maximum speed (km/h) a motorist may have without leaving the road at the crest of the hill?

7) An airplane is flying in a horizontal circle at 480 km/h. The wings are tilted at 40o with respect to the horizontal. Assume that the lift force of the wings is directed perpendicular to the wings. What is the radius of the circle in which the airplane is flying?

8) Assume for a moment that you are traveling around a loop in the vertical plane at a constant velocity of 20 m/s. What is the function that describes the radius of the loop at each point (a function of θ makes a lot of sense) so that the normal force you experience is a constant 2 g's? (Give this a try - it does have an answer and its not as bad as you might think:) Here's a picture of a similar loop for you to consider. This picture was taken from a Gizmodo article that you may find interesting.

9) Assume that you are traveling along a path which forms a logarithmic spiral (radius of curvature at any point is given by r=ea θ where θ is your angular position as a function of time). Determine an expression for your speed at any angular position θ if the net radial force is "1g". Now, using concepts from integral calculus, determine the angular position θ as a function of time. From this you should now be able to write down your linear speed as a function of time. What does the derivative of this expression tell you? At this stage of your learning this is a challenging problem because of the need to understand the underlying concepts clearly and not be thrown off by the strange symbols.