# Soluções Haliday - Fisica II

(Parte 1 de 3)

Chapter 12

1. (a) The center of mass is given by m m m x

(b) Similarly, we have

6 m m m my m

(c) Using Eq. 12-14 and noting that the gravitational effects are different at the different locations in this problem, we have i i i i

i i i xm g x m g x m g x m g x m g x m g x m g x m g m g m g m g m g m gm g

2. The situation is somewhat similar to that depicted for problem 10 (see the figure that accompanies that problem). By analyzing the forces at the “kink” where ρ F is exerted, we find (since the acceleration is zero) 2T sin θ = F, where θ is the angle (taken positive) between each segment of the string and its “relaxed” position (when the two segments are collinear). Setting T = F therefore yields θ = 30º. Since α = 180º – 2θ is the angle between the two segments, then we find α = 120º.

3. The object exerts a downward force of magnitude F = 3160 N at the midpoint of the rope, causing a “kink” similar to that shown for problem 10 (see the figure that accompanies that problem). By analyzing the forces at the “kink” where ρ F is exerted, we find (since the acceleration is zero) 2T sinθ = F, where θ is the angle (taken positive) between each segment of the string and its “relaxed” position (when the two segments are colinear). In this problem, we have

4. From ρ ρ ρ τ=×rF, we note that persons 1 through 4 exert torques pointing out of the page (relative to the fulcrum), and persons 5 through 8 exert torques pointing into the page.

(a) Among persons 1 through 4, the largest magnitude of torque is (330 N)(3 m) = 990 N·m, due to the weight of person 2.

(b) Among persons 5 through 8, the largest magnitude of torque is (330 N)(3 m) = 990 N·m, due to the weight of person 7.

5. Three forces act on the sphere: the tension force ρ T of the rope

(acting along the rope), the force of the wall NFr (acting horizontally away from the wall), and the force of gravity mgρ (acting downward). Since the sphere is in equilibrium they sum to zero. Let θ be the angle between the rope and the vertical. Then Newton’s second law gives

 vertical component : T cos θ – mg = 0 horizontal component: FN – T sin θ = 0.

(a) We solve the first equation for the tension: T = mg/ cos θ. We substitute cosθ=+LLr/22to obtain

(b) We solve the second equation for the normal force: sinNFTθ=.

Usingsinθ = +r L r/ 2 2 , we obtain

Tr mg L r r mgrF

L L r L r

6. Our notation is as follows: M = 1360 kg is the mass of the automobile; L = 3.05 m is the horizontal distance between the axles; (3.051.78) m1.27 m=−=lis the horizontal distance from the rear axle to the center of mass; F1 is the force exerted on each front wheel; and, F2 is the force exerted on each back wheel.

(a) Taking torques about the rear axle, we find

(b) Equilibrium of forces leads to1222,FFMg+=from which we obtainF2338910=×.N.

7. We take the force of the left pedestal to be F1 at x = 0, where the x axis is along the diving board. We take the force of the right pedestal to be F2 and denote its position as x = d. W is the weight of the diver, located at x = L. The following two equations result from setting the sum of forces equal to zero (with upwards positive), and the sum of torques (about x2) equal to zero:

+ − = (a) The second equation gives

L dF W d

which should be rounded off to 311.210 NF=−×. Thus, 31||1.210 N.F=× (b) Since F1 is negative, indicating that this force is downward.

which should be rounded off to 321.710 NF=×. Thus, 32||1.710 N.F=× (d) The result is positive, indicating that this force is upward.

(e) The force of the diving board on the left pedestal is upward (opposite to the force of the pedestal on the diving board), so this pedestal is being stretched.

(f) The force of the diving board on the right pedestal is downward, so this pedestal is being compressed.

8. Let λ115=.mand2(5.01.5) m3.5 m=−=l. We denote tension in the cable closer to the window as F1 and that in the other cable as F2. The force of gravity on the scaffold itself (of magnitude msg) is at its midpoint, λ325=.mfrom either end.

(a) Taking torques about the end of the plank farthest from the window washer, we find w sm g m gF + = =

l l l l

(b) Equilibrium of forces leads to which (using our result from part (a)) yieldsF225310=×.N.

9. The forces on the ladder are shown in the diagram on the right. F1 is the force of the window, horizontal because the window is frictionless.

F2 and F3 are components of the force of the ground on the ladder. M is the mass of the window cleaner and m is the mass of the ladder.

The force of gravity on the man acts at a point 3.0 m up the ladder and the force of gravity on the ladder acts at the center of the ladder. Let θ

be the angle between the ladder and the ground. We use

22cos/ or sin/ dLLdLθθ==−to find θ = 60º. Here L is the length of the ladder (5.0 m) and d is the distance from the wall to the foot of the ladder (2.5 m).

(a) Since the ladder is in equilibrium the sum of the torques about its foot (or any other point) vanishes. Let lbe the distance from the foot of the ladder to the position of the window cleaner. Then,

and 2

M mL gF L θθ

This force is outward, away from the wall. The force of the ladder on the window has the same magnitude but is in the opposite direction: it is approximately 280 N, inward.

(b) The sum of the horizontal forces and the sum of the vertical forces also vanish:

The first of these equations gives FF3122810==×.Nand the second gives

The magnitude of the force of the ground on the ladder is given by the square root of the sum of the squares of its components:

 2 2 2 228 10 83 10 8 10( . ( N) N) N.

(c) The angle φ between the force and the horizontal is given by so φ = 71º. The force points to the left and upward, 71º above the horizontal. We note that this force is not directed along the ladder.

10. The angle of each half of the rope, measured from the dashed line, is

Analyzing forces at the “kink” (where ρ F is exerted) we find

1. The x axis is along the meter stick, with the origin at the zero position on the scale. The forces acting on it are shown on the diagram below. The nickels are at x = x1 = 0.120 m, and m is their total mass. The knife edge is at x = x2 = 0.455 m and exerts force ρ F. The mass of the meter stick is M, and the

force of gravity acts at the center of the stick, x = x3 = 0.500 m.

Since the meter stick is in equilibrium, the sum of the torques about x2 must vanish:

(Parte 1 de 3)