CHAPTER 31
31.21
31.22
31.23
A 100-ft cable weighing 5 Ib/ft supports a safe weighing 500 Ib. Find the work done in winding 80 ft of the cable
on a drum.
Let x denote the length of cable that has been wound on the drum. The total weight (unwound cable and
safe) is 500 + 5(100-*) = 1000- 5x, and the work done in raising the safe a distance A* is (1000-5x) Ax.
Hence, the work W= J 0
80 (1000 - 5x) dx = (1000* - fx
2 ) ]*° = 64,000 ft • Ib.
A particle carrying a positive electrical charge + q is released at a distance d from an immovable positive charge
+ Q. How much work is done on the particle as its distance increases to 2d?
By Coulomb's law, the repulsive force on the particle when at distance d + x is kQql(d + x)
2 , where k is a
universal constant. Then the work is
The expansion of a gas in a cylinder causes a piston to move so that the volume of the enclosed gas increases from
15 to 25 cubic inches. Assuming the relationship between the pressure p (lb/in
2 ) and the volume y (in
3 ) is
pv
1 '
4 = 60, find the work done.
If A is the area of a cross section of the cylinder, pA is the force exerted by the gas. A volume increase Ay
causes the piston to move a distance AuM, and the corresponding work is
CENTROID OF A PLANAR REGION
In Problems 31.24-31.32, find the centroid of the given planar region.
31.24 The region bounded by v = x
2 , y= 0, and * = 1 (Fig. 31-3).
Fig. 31-3
256
See Fig. 31-2. Let x be the distance from the bottom of the tank. Consider a slab of water of thickness Ax at
depth*. By similar triangles, we see that r/x=-fa, where r is the radius of the slab. Then r = 2x/5. The
weight of the slab is approximately wirr
2 Ax, where w is the weight-density of water (about 9.8kN/m
3 ) and
the slab is raised a distance of 10 — x. Hence, the work is
Then the work
The area
The moment about the y-axis is
The moment about the x-axis is
Hence, the x-coordinate of the centroid is
Hence, the y-coordinate of the centroid is y=
Thus, the centroid is
31.25
The region bounded by the semicircle y —
and y = 0
By symmetry,
(In general, the centroid lies on any line of symmetry of the region.) The area
The moment about the x-axis
Thus, the centroid is
Thus,
x = 0 .
31.21
31.22
31.23
A 100-ft cable weighing 5 Ib/ft supports a safe weighing 500 Ib. Find the work done in winding 80 ft of the cable
on a drum.
Let x denote the length of cable that has been wound on the drum. The total weight (unwound cable and
safe) is 500 + 5(100-*) = 1000- 5x, and the work done in raising the safe a distance A* is (1000-5x) Ax.
Hence, the work W= J 0
80 (1000 - 5x) dx = (1000* - fx
2 ) ]*° = 64,000 ft • Ib.
A particle carrying a positive electrical charge + q is released at a distance d from an immovable positive charge
+ Q. How much work is done on the particle as its distance increases to 2d?
By Coulomb's law, the repulsive force on the particle when at distance d + x is kQql(d + x)
2 , where k is a
universal constant. Then the work is
The expansion of a gas in a cylinder causes a piston to move so that the volume of the enclosed gas increases from
15 to 25 cubic inches. Assuming the relationship between the pressure p (lb/in
2 ) and the volume y (in
3 ) is
pv
1 '
4 = 60, find the work done.
If A is the area of a cross section of the cylinder, pA is the force exerted by the gas. A volume increase Ay
causes the piston to move a distance AuM, and the corresponding work is
CENTROID OF A PLANAR REGION
In Problems 31.24-31.32, find the centroid of the given planar region.
31.24 The region bounded by v = x
2 , y= 0, and * = 1 (Fig. 31-3).
Fig. 31-3
256
See Fig. 31-2. Let x be the distance from the bottom of the tank. Consider a slab of water of thickness Ax at
depth*. By similar triangles, we see that r/x=-fa, where r is the radius of the slab. Then r = 2x/5. The
weight of the slab is approximately wirr
2 Ax, where w is the weight-density of water (about 9.8kN/m
3 ) and
the slab is raised a distance of 10 — x. Hence, the work is
Then the work
The area
The moment about the y-axis is
The moment about the x-axis is
Hence, the x-coordinate of the centroid is
Hence, the y-coordinate of the centroid is y=
Thus, the centroid is
31.25
The region bounded by the semicircle y —
and y = 0
By symmetry,
(In general, the centroid lies on any line of symmetry of the region.) The area
The moment about the x-axis
Thus, the centroid is
Thus,
x = 0 .