Examples

 

1.  Solution for the donut exercise: 9 problems

 

2.  Sometimes people say that water is removed from clothes in a spin dryer by centrifugal force throwing the water outward.

      What is wrong with this statement?

 

There is nothing to cause an outward force. The drum pushes inward on the water. The water's velocity vector is tangent to the circle; it escapes through holes in the drum and continues on its tangential path.

 

 

3.  Will the acceleration of a car be the same when the car travels around a sharp curve at a constant speed as when it travels around

     a gentle curve (larger radius) at the same speed (the magnitude of the velocity does not change)?

 

a_r = v²/r

 

As r increases, the radial acceleration decreases. For a more gentle curve (larger r) the acceleration would be less

 

4.  Why do airplanes bank when they turn?

 

The airplane is able to fly because of Bernoulli's principle (chapter 10) which states that as the velocity of a fluid goes up the pressure goes down and vice versa. Because of the curvature of the wing, the velocity of air on the bottom decreases so pressure upwards increases.

Draw the free body diagram for the airplane. There is a force downward due to the airplane's mass (F = ma) and an upward normal force. See above.

In order to turn, there must be a non-vertical component of the normal force which points toward the center of the curve, providing the centripetal force.

Note that the sum of the vertical forces must be 0 for the plane to execute a level turn.

This results in F_liftcos q = mg.

The horizontal component of the lifting force provides the centripetal force to move the plane in a circle.

F_lift sin q = mv²/r.

The above equation can be solved to find the correct angle for the turn.

 

5.  A flat puck of mass M1 rotates in a circle on a frictionless air-hockey tabletop. It is held in this orbit by a light cord connected to a

     dangling block of mass M2.  Determine an expression for the speed of the puck.

 

Draw the 2 free body diagrams; one for the puck, M1, and one for the mass M2. Pick positive as up.

Use Newton's law to sum the forces for each mass.

The sum in the vertical direction for mass 1 must = 0. The sum in the horizontal direction for mass 1 = m1a_r = m1rv²/r. Set this equal to the tension force in the cord.

For mass 2 a sum in the vertical direction gives the tension force - from above - equal to the force of gravity exerted on the second mass.

 

6.  How many revolutions per minute would a Ferris wheel of a given diameter have to make for the passengers at the top to feel

      weightless?

 

Draw the free body diagram with the normal force pointing up (choose up as positive) and the force of gravity on the passengers pointing down.

Note that for a feeling of weightlessness, the normal force must = 0.

Set mg = ma_r and solve for v

Once you have v, you can then use the diameter of the circle to solve for the RPM

 

7. What is the magnitude of the gravitational force between two objects, one weighing 80 kg and the other weighing 95 kg?

     Assume the 2 objects are separated by a distance of 2.0 meters.

 

We use Newton's law of universal gravitation

 

F = G(m₁*m₂)/r²  = (10^-10 N m²/kg²)(80 kg)((95 kg)/(2.0 m)² = 7.6 x 10^-7 N

 

8. Road friction problems discussed in the text.

 

These are good examples of practical applications of this materal. Be sure you understand the approach.

 

Road friction required so that car can travel around the curve without slipping

Choose up as positive and draw the free body diagram for the car.

Since there is no acceleration in the vertical direction, the normal force = mg, solve for the normal force

Sum forces in the radial direction.

You can then obtain an expression for the static friction required in terms of the other variables

 

Banking angle to use so that no friction is required

Draw the free body diagram with positive as up, along a normal from the road surface.

Draw a wedge to simulate the banking angle - insert the forces and the angle (recall the problems I solved earlier and you did in class)

Place the normal force acting normal to the surface.

Place the force of gravity acting on the car pointing down

Use Newton's law and sum the forces in the x and y directions

The sum in the y direction is 0

Solve for the angle