Page 264 - Fisika Terapan for Engineers and Scientists

464 CHAPTER 14 Statics and Elasticity

104. A rope of length 12 m consists of an upper half of nylon of 105. Suppose you drop an aluminum sphere of radius 10 cm into
diameter 1.9 cm spliced to a lower half of steel of diameter the ocean and it sinks to a depth of 5000 m, where the pres-
7
2
0.95 cm. How much will this rope stretch if a mass of 4000 kg sure is 5.7 10 N/m . Calculate by how much the diameter
is suspended from it? The Young’s modulus for steel rope is of this sphere will shrink.
10
2
19 10 N/m .

Answers to Checkups

Checkup 14.1 bottom that balances the torque due to the other’s weight.
Such a torque increases from zero when the pieces of lumber
1. If the bicyclist sits rigidly, the equilibrium is unstable: if tipped are vertical (when the tip of the A makes zero angle) to a max-
slightly, gravity will pull the bicycle and cyclist further over. imum when the tip approaches 180 . Since the force exerted
2. When you extend your legs while sitting on a swing, you are by one piece on the other acts with a smaller and smaller
shifting your center of mass forward. To remain in equilib- moment arm as the tip angle approaches 180 , the force must
rium, the swing and your body will shift backward, and tilt, so be very large as the tip angle approaches 180 .
as to keep your center of mass aligned below the point of sup- 4. (B) Increase the upward push and increase the downward
port. push. If we consider the torques about an axis through the for-
3. (a) Yes, the (vertical) support force is along the same line as ward hand, then the downward pull from the fish must be bal-
the weight when holding the pole straight up (more precisely, anced by increasing the downward push from the rear hand.
it is slightly distributed around the edge of the pole). (b) No, The upward push of the forward hand must increase to bal-
the support force is provided by the more forward hand ance those two increased downward forces.
(which pushes up); an additional force from the rear hand
pushes down, to balance the torques from the force of gravity Checkup 14.3
and the support force.
1. The arrangement shown in Fig. 14.23b has the larger ratio
4. (D) Neutral, stable, unstable. As our intuition might suggest, a
l l , and thus has a greater mechanical advantage.
cone on its side is in neutral equilibrium (after a small dis-
placement, it remains on its side). A cone on its base is in 2. No. If, for example, the force F is not perpendicular to the
stable equilibrium (after being tipped slightly, it will settle lever, we must replace l by l sin , where is the angle between
back on its base). Finally, a cone balanced on its apex is in the force and the lever.
unstable equilibrium (after being tipped slightly, the cone will 3. No. The pulley transmits tension to a different direction, inde-
fall over). pendent of its size.
4. (C) 100 N. The weight of the rock is w mg 100 kg
2
Checkup 14.2 9.8 m/s 980 N 1000 N. The lever has a mechanical
1
advantage of l l . So the force required to lift the rock is
10
1. When a ladder makes a large angle with the vertical, the F (l >l)F 10 1 1000 N 100 N .
weight of the ladder and the person climbing it exerts a large
torque about the bottom, which can more easily overcome Checkup 14.4
friction and make the ladder slip. When a ladder makes a
small angle with the vertical, a person on the ladder can shift 1. The tension determines the fractional elongation [see Eq.
the center of mass to a point behind the bottom, causing the (14.18)]; thus, for a piano wire of twice the length, the elonga-
ladder to topple backward. tion will be twice as long, or 4.0 mm.
2. With the center of mass on the bottom, the box would have to 2. Yes, a cable can snap under its own weight (the downward
be rotated 90 before toppling over. In that case, however, the weight below any point must be balanced by the upward ten-
box would then be on its side when it reaches the critical sion at that point). Since the critical length for breaking is a
angle, where the center of mass is just above the support point. condition of maximum tensile stress (a force per unit area),
3. For each side of the A, the force that one piece of lumber this depends on only the material and its mass density, not its
exerts on the other must exert a torque about the other’s area.