GENERAL PHYSICS LECTURE

Chapter 6

CIRCULAR MOTION AND

OTHER APPLICATIONS

OF NEWTON’S LAWS PhD: Nguyễn Thị Ngọc Nữ

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

6.1 Newton’s Second Law Applied to Uniform Circular Motion

6.2 Nonuniform Circular Motion

6.3 Motion in Accelerated Frames

6.1 Newton’s Second Law Applied to Uniform Circular Motion

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

2vcF ma m

r

Quick Quiz

You are riding on a Ferris wheel (Fig. 6.3) that

is rotating with constant speed. The car in

which you are riding always maintains its

correct upward orientation—it does not

invert. What is the direction of your

centripetal acceleration when you are at the

top of the wheel? (a) upward (b) downward

(c) impossible to determine. What is the

direction of your centripetal acceleration

when you are at the bottom of the wheel?

Quick Quiz

Fig. 6.3

6.1 Newton’s Second Law Applied to Uniform Circular Motion

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

Example 1

A small object of mass m is suspended from

a string of length L. The object revolves with

constant speed v in a horizontal circle of

radius r, as shown in Figure 6.4. (Because

the string sweeps out the surface of a cone,

the system is known as a conical pendulum.)

Find an expression for v.

6.1 Newton’s Second Law Applied to Uniform Circular Motion

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

Example 1

6.1 Newton’s Second Law Applied to Uniform Circular Motion

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

Example 2

A 1 500-kg car moving on a flat, horizontal

road negotiates a curve, as shown in Figure

6.5. If the radius of the curve is 35.0 m and

the coefficient of static friction between the

tires and dry pavement is 0.500, find the

maximum speed the car can have and still

make the turn successfully.

6.1 Newton’s Second Law Applied to Uniform Circular Motion

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

Example 2

6.1 Newton’s Second Law Applied to Uniform Circular Motion

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

Example 3

A pilot of mass m in a jet aircraft executes a

loop-the-loop, as shown in Figure 6.7a. In

this maneuver, the aircraft moves in a

vertical circle of radius 2.70 km at a

constant

speed of 225 m/s. Determine the force

exerted by the seat on the pilot (A) at the

bottom of the loop and (B) at the top of the

loop.

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

Example 3

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

Example 3

6.1 Newton’s Second Law Applied to Uniform Circular Motion

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

Problem

A 4.00-kg object is attached to a vertical

rod by two strings, as in Figure P6.11. The

object rotates in a horizontal circle at

constant speed 6.00 m/s. Find the tension

in (a) the upper string and (b) the lower

string.

6.1 Newton’s Second Law Applied to Uniform Circular Motion

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

Problem

Figure P6.11

6.2 Nonuniform Circular Motion

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

r ta a a

r tF F F

6.2 Nonuniform Circular Motion

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

Quick Quiz 1

Which of the following is impossible for a car

moving in a circular path?

(a)the car has tangential acceleration but no

centripetal acceleration.

(b)the car has centripetal acceleration but no

tangential acceleration.

(c)the car has both centripetal acceleration

and tangential acceleration.

6.2 Nonuniform Circular Motion

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

Quick Quiz 2A bead slides freely along a horizontal,

curved wire at constant speed, as shown in

Figure. Draw the vectors representing the

force exerted by the wire on the bead at

points A, B, and C.

6.2 Nonuniform Circular Motion

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

Quick Quiz 3In Figure, the bead speeds up with constant

tangential acceleration as it moves toward

the right. Draw the vectors representing the

force on the bead at points A, B, and C.

6.2 Nonuniform Circular Motion

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

A small sphere of mass m is attached to the

end of a cord of length R and set into motion

in a vertical circle about a fixed point O, as

illustrated in Figure 6.10. Determine the

tension in the cord at any instant when the

speed of the sphere is v and the cord makes

an angle q with the vertical.

Example

6.2 Nonuniform Circular Motion

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

Example

6.2 Nonuniform Circular Motion

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

A pail of water is rotated in a vertical

circle of radius 1.00 m. What is the

minimum speed of the pail at the top of

the circle if no water is to spill out?

Problem

6.2 Nonuniform Circular Motion

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

Problem

6.3 Motion in Accelerated Frames

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

a

6.3 Motion in Accelerated Frames

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS

6.3 Motion in Accelerated Frames

CIRCULAR MOTION AND OTHER APPLICATIONS OF NEWTON’S LAWS