I. Warm-up Exercises
1. A bee circles a hive while waiting to gain entrance to the main honey-storage compartment. He moves at 0.32 m/s and is found to take 48 s to completely encircle the hive 5 times. (a) What is the period of the bee’s circular motion? (b) What is the linear frequency of the bee’s motion? (c) What is its angular frequency? (d) What is the radius of the circular motion about the hive?
II. Some Standards
2. An old record is spinning on an equally old turntable at 78 rpm (revolutions per minute). Answer all of the questions below in MKS units. (a) How many times does the record rotate per second? (b) What is the angular frequency of the record? (c) How long does it take the record to complete one revolution? (d) What is the speed of a dust particle stuck to the record at a point which is 4 inches from the record’s axis of rotation? (e) What is the centripetal acceleration of the dust particle mentioned above?
3. A 400-kg plane is flying in a horizontal circle of radius 1.3 km. In order to make the turn, the plane is banking at an angle of 20o. (That is, it is tilted from its horizontal position by 20 o.) The lift force L provided by the wings acts as a kind of normal force which acts in the direction shown below (perpendicular to the wings). Ignore any frictional effects in this problem.
(a) What is the magnitude of the lift force provided by the plane’s wings? (Hint: Look at the sum of forces in the vertical direction. What is the acceleration of the plane in the vertical direction? Is it even moving in the vertical direction?) (b) With what constant speed must the plane be flying? (Hint: What’s the equation for the centripetal acceleration? Look at Newton’s 2nd law applied to the c-direction!)
III. So, you think you’re pretty good...?
4. A 73-kg pilot decides to test his knowledge of physics. He places a bathroom scale on his seat in the plane, sits in his seat on the bathroom scale, notices that his weight looks normal as read off of the scale, and then takes off. During one part of his flight, the pilot swoops down in a vertical circular arc of radius 60 m. (Imagine a vertical circle in the air. The pilot is flying along the bottom portion of this circle, curving down, reaching the bottom, and then curving back up.) With what speed must the plane be traveling at the bottom of the arc if the scale reads twice his weight on the ground? (Hint: Draw a FBD for the pilot!)
5. A hole is drilled in the center of an air-hockey table and a string is passed through the hole. One end of the string is attached to an air-hockey puck which can glide frictionlessly on a cushion of air on top of the table. A mass m is hung from the other end of the string which hangs freely under the table, as shown in the figure below. After playing around with the puck, you find that you can get the puck moving around in a circle (centered on the hole) of radius 25 cm if it is moving with a speed of 2.1 m/s. (a) What is the puck’s period of motion? (b) What is the angular velocity of the puck? (c) Find the value of the mass m if the mass of the puck is 120 g. (Hint: You will need to draw two free-body diagrams — one for the puck, and one for the hanging mass. What is the tension in the string? In what direction does this string tension act on the puck?)
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