**Vertical Circular Motion Problems**1. A ball with a mass of 130 g is swung at the end of a string 93.0 cm in length. The ball is whirled in a vertical circle at 4.00 revolutions per second.

a. What is the tension on the string at the bottom of the loop?

b. What is the tension on the string at the top of the loop?

2. A jet fighter pilot knows he is able to withstand an acceleration of 9g before blacking out. The pilot points his plane vertically down while traveling at Mach 3 speed and intends to pull up in a circular maneuver before crashing to the ground.

a) Where does the maximum acceleration occur in the maneuver?

b) What is the minimum radius the pilot can take?

3. A roller coaster at an amusement park has a dip that bottoms out in a vertical circle of radius r. A passenger feels the seat of the car pushing upward on her with a force equal to 3.0 times her weight as she goes through the dip. If r = 25.0 m, how fast is the roller coaster traveling at the bottom of the dip?

4. What is the apparent weight of a 75-kg person driving a car with a constant speed of 12 m/s over a bump with a circular cross-section and radius of curvature of of 35 m?

5. What is the minimum speed of a roller coaster at the top of a 39.0 m vertical loop if the passengers are "weightless" at that point.

6. A ball of a mass 4.0 kg is attached to the end of a 1.2 cm long string and whirled around in a circle that describes a vertical plane.

a. What is the minimum speed that the ball can be moving at and still maintain a circular path? Provide a free body diagram.

b. At this speed, what is the maximum tension in the string? Provide a free body diagram.

c. If the ball was swung in a horizontal circle at this speed, what angle would the string make with the vertical?

7. How do you find the tension in the string of a ball traveling in a vertical circle at the 45 degree angle?

8. A hill is in the shape of an arch having the radius of curvature of 41. m. What is the maximum speed that a car can travel across the hill without 'getting some air'?

**Banked Curves**1. Determine the minimum angle at which a road should be banked so that a car traveling at 20.0 m/s can safely negotiate the curve if the radius of the curve is 200.0 m.

2. If a curve with a radius of 65 m is properly banked for a car traveling 75 km/h, what must be the coefficient of static friction for a car not to skid when traveling at 90 km/h?

3. A Car is driven around a circle with a radius of 200m, bank angle 10 degrees. The static frictional coefficient is 0.60. Calculate the maximum velocity the car can travel.

4. An airplane is flying in a horizontal circle at a speed of 460 km/h. If its wings are tilted 40° to the horizontal, and force is provided by lift that is perpendicular to the wing surface. What is the radius of the circle?

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