ch-5

F f_k = _kn fs = F 0 | f| fs,max Static region (c) (a) (b) Kinetic region µ mg n F n Motion mg fk fs F Figure 5.1

(a ) Th e force of static friction between a trash can and a concrete patio is opposite the applied force . Th e magnitude of the force of static friction equals that of the applied force. (b ) When the magnitude of the applied force exceeds the magnitude of the force of kin etic friction , the trash can accelerates to the right. (c ) A graph of the magnitude of the friction force versus that of the applied force. In our model, the force of kin etic friction is independent of the applied force and the relative speed of the surfaces. Note that

fs,max fk. f : k F : f : s

More Applications of Newton’ s Laws

Chapter 5

30°

F _30°

(a) (b)

F

F

IGURE

5.2

(Quick Quiz 5.3) A father tries to slide his daughter on a sled over snow by (a) pushing downward on her shoulders or (b) pulling upward on a rope attached to the sled. Which is easier?

Gantry Bow of ship "Umbilical cord" Oil rig (a) (b) Abyss

"Umbilical cord" is almost horizontal

Oil rig moves toward abyss

(c) "Umbilical cord" is almost vertical

Abyss Oil rig stops short of edge

fk mg n g n f

F

I G U R E

5.5

(Example 5.2) A block on an adjustable incline is used to determine the coefficients of friction.

n f y x θ mg sin mg cos θ mg θ θ

F

I G U R E

5.6

(Example 5.3) (a) Two objects connected by a light string that passes over a frictionless pulley. (b) Free-body diagram for the sliding cube. (c) Free-Free-body diagram for the hanging ball.

(c) T 7.0 kg m2g 4.0 kg 7.0 kg (a) T 4.0 kg m1g fk n (b)

F

I G U R E

5.7

(Example 5.4) (a) A crate of mass m slides down an incline. (b) Free-body diagram for the sliding crate.

d θ (a) a (b) y x f_k θ mg cosθ mg sinθ n mg

F

I G U R E

5.8

Overhead view of a ball moving in a circular path in a horizontal plane. A force directed toward the center of the circle keeps the ball moving in its circular path. F : r m Fr F_r r

r

Figure 5.9

An overhead view of a ball moving in a circular path in a horizontal plane. When the string breaks, the ball moves in the direction tangent to the circular path.

F

I G U R E

5.10

(Quick Quiz 5.4) A Ferris wheel located on Navy Pier in Chicago, Illinois.

(©

F

I G U R E

5.11

(Example 5.6) The conical pendulum and its free-body diagram.

T mg T cos mg T sin r θ _θ θ θ L

n mg (a) (b) f_s f_s

F

I G U R E

5.12

(Interactive Example 5.7) (a) The force of static friction directed toward the center of the curve keeps the car moving in a circular path. (b) Free-body diagram for the car.

F

I G U R E

5.13

(Interactive Example 5.8) A car rounding a curve on a road banked at an angle to the horizontal. In the absence of friction the force that causes the centripetal acceleration and keeps the car moving in its circular path is the horizontal component of the normal force.

n ny

F_g

θ θ

nbot mg ntop mg (b) (c) Top Bottom (a)

F

I G U R E

5.14

(Example 5.9) (a) An aircraft executes a loop-the-loop maneuver as it moves in a vertical circle at constant speed. (b) Free-body diagram for the pilot at the bottom of the loop. In this position, the pilot experiences a force from the seat that is larger than his weight. (c) Free-body diagram for the pilot at the top of the loop. Here the force from the seat could be smaller than his weight or larger, depending on the speed of the aircraft.

F

I G U R E

5.16

(Quick Quiz 5.6) A bead slides along a curved wire.

F F_r

F_t F Figure 5.15

When the net force acting on a particle moving in a circular path has a tangential component vector , its speed changes. The total force on the particle also has a component vector directed toward the center of the circular path. Therefore, the total force is F: F:_r F:_t.

F:_r

F

I G U R E

5.17

(Example 5.10) (a) Forces acting on a sphere of mass m connected to a cord of length R and rotating in a vertical circle centered at O. (b) Forces acting on the sphere when it is at the top and bottom of the circle. The tension has its maximum value at the bottom and its minimum value at the top.

O Tbot Ttop vbot mg mg vtop (b) (a) R O T _θ mg cos mg sin mg θ θ θ

(b) v vT 0.632vT t R mg v (a) Figure 5.18

(a) A small sphere falling through a viscous fluid. (b) The speed–time graph for an object falling through a viscous medium. The object approaches a terminal speed vT, and

the time constant is the time interval required to reach 0.632vT.

v v_T R mg R mg

F

I G U R E

5.19

An object falling through air experiences a resistive drag force and a gravitational force . The object reaches terminal speed (on the right) when the net force acting on it is zero, that is, when , or R mg. Before that

occurs, the acceleration varies with speed according to Equation 5.9.

R : F : g F : g mg: R :

F

I G U R E

5.20

(Quick Quiz 5.7) A sky surfer takes advantage of the

F

I G U R E

5.22

Two point charges separated by a distance r exert an electrostatic force on each other given by Coulomb’s law. (a) When the charges are of the same sign, the charges repel each other. (b) When the charges are of opposite sign, the charges attract each other.

– + r (a) F_e F_e q₁ q₂ (b) Fe F_e q₁ q2 + +

F

I G U R E

5.21

Two particles with masses m1and m2attract each other with a force of magnitude Gm1m2/r2.

r

–Fg F_g

m₂ m1

θ Figure P5.8 5.00 kg 9.00 kg Figure P5.10 22.0° 22.0° Figure P5.4 Figure P5.3

θ Figure P5.19 F m₂ T m1 Figure P5.11 1.00 kg 2.00 kg 4.00 kg Figure P 50.0° Figure P5.13

3.00 m 2.00 m 2.00 m Figure P5.20 v Figure P5.22 Figure P5.24

F_r Ft F F_r F F_t Ft (a) (b) Figure QQA.5.6