|
Centripetal Force & Gravity Centripetal Force Centripetal Acceleration – A center-seeking force must be exerted on an object if it is to move in a curved path. There is a proof in the book on centripetal acceleration. The formula to calculate the centripetal acceleration is:
To calculate the force of centripetal acceleration you use Newton's Second Law:
Some Uniform Circular Motion Examples Example 1 The bobsled track at the 1994 Olympics in Lillehammer, Norway, contained turns with radii of 33 m and 24 m, as in the figure. Find the centripetal acceleration at each turn for a speed of 34 ms, a speed that was achieved in the two-man event. Express the answers as multiples of g.
Example 2 The model airplane in the figure has a mass of 0.90 kg and moves at a constant speed on a circle that is parallel to the ground. The path of the airplane and its guideline lie in the same horizontal plane, because the weight of the plane is balanced by the lift generated by its wings. Find the tension T in the guideline (length = 17 m) for speeds of 19 m/s and 38 m/s.
Example 3 A 1200 kg car
rounds a corner of radius r = 45 m. If the coefficient of static friction
between the tires and the road is
Example 4 The figure represents
a stock car of mass m = 1600 kg traveling at a constant speed v = 20
m/s around a flat, circular track of radius R = 190 m. For what value
of
Banked Curves When a car travels around a corner friction is the force that is accelerating the car around the corner. To eliminate the need for friction the road can be banked. Then we find the force that is directed along the radius of the curve, as in the figure below.
Then the vertical force is the weight.
We can then divide the two equations and simplify.
Example 5 If a roadway is banked at the proper angle, a car can round a corner without any assistance from friction between the tires and the road. Find the appropriate angle for a 900 kg car traveling at 20.5 m/s in a turn of radius 85.0 m.
Example 6 While driving along a country lane with a constant speed of 17.0 m/s, you encounter a dip in the road. The dip can be approximated as a circular arc, with radius of 65.0 m. What is the normal force exerted by a car seat on a 80.0 kg passenger when the car is at the bottom of the dip?
Example 7 The centrifuge in the figure rotates at a rate that gives the bottom of the test tube a linear speed of 89.3 m/s. If the bottom of the test tube is 8.50 cm from the axis of rotation, what is the centripetal acceleration experienced there?
Example 8 In a 1901 circus performance, Allo "Dare Devil" Diavolo introduced the stunt of riding a bicycle in a loop-the-loop. Assuming that the loop is a circle with radius R = 2.7 m, what is the least speed v Diavolo could have at the top of the loop to remain in contact with it there?
Example 9 The figure shows
a conical pendulum, in which the bob moves in a horizontal circle at
constant speed. (The cone sweeps out a cone as the bob rotates.) The
distance L between the top of the cord and the center of the bob is
1.7 m. The cord makes an angle of
|
,
what is the greatest speed the car can have in the corner without skidding?
between
the track and the tires of the car will the car be on the verge of
sliding off the track?
with
the vertical. Find the period 
of
the motion.