第6.6节 某些物理学方面的应用

6.6 SOME APPLICATIONS TO PHYSICS

The Infinite Sum Theorem can frequently be used to derive formulas in physics.

1 MASS AND DENSITY, ONE DIMENSION

Consider a one- dimensional object such as a length of wire. We ignore the atomic nature of matter and assume that it is distributed continuously along a line segment. If the density ρ per unit length is the same at each point of the wire, then the mass is the product of the density and the length, m= ρL. If L is in centimeters and ρ in grams per centimeter, then m is grams. ( ρ is the Greek letter “ rho”.)

Now suppose that the density of the wire varies continuously with the position. Put the wire on the x-axis between the points x=a and x=b, and let the density at the point x be ρ(x). Consider the piece of the wire of infinitesimal length Δx and mass Δm shown in Figure 6.6.1. At each point between x and x+ Δx, the density is infinitely close to ρ(x), so

Δm ≈ ρ (x)Δx (compared to Δx ).

Figure 6.6.1

Therefore by the Infinite Sun Theorem, the total mass is

m = _________ ρ(x) dx.

EXAMPLE 1

Find the mass of a wire 6 cm long whose density at distance x from the center is 9 - x² gm/cm. In Figure 6.6.2, we put the center of the wire at the origin. Then

m = ___ 9 - x²dx = 9x - ____ x3] ______ = 36 gm.

Figure 6.6.2

2 MASS AND DENSITY, TWO DIMENSIONS

Imagine a flat plate which occupies the region below the curve y= f(x), f(x) ≥ 0, from x=a to x=b. If its density per unit area is a constant ρ gm/cm², then its mass is the product of the density and area.

m= ρ A=ρ ______f(x) dx.

Suppose instead that the density depends on the value of x, ρ(x). Consider a vertical strip of the plate of infinitesimal width Δx (Figure 6.6.3). On the strip between x and x+ Δx, the density is everywhere infinitely close to ρ(x), so

Δm ≈ ρ (x) ΔA≈ρ(x)f(x)Δx. (compared to Δx).

Figure 6.6.3

By the Infinite Sum Theorem,

m= _____ ρ(x)f(x)dx.

EXAMPLE 2

A circular disc of radius r has density at each point equal to the distance of the point from the y-axis. Find its mass. ( the center of the circle, shown in Figure 6.6.4, is at the origin.) the circle is the region between the curves __________ from -r to r. The density at a point (x, y) in the disc is | x|. By symmetry, all four quadrants have the same mass. We shall find the mass m1 of the first quadrant and multiply by four.

Put u= r² - x², du = -2x dx; u= r² when x=0, and u=0 when x=r.

Then m = 4m1 = ___ r 3.

Figure 6.6.4

3 MOMENTS, ONE DIMENSION

Two children on a weightless seesaw will balance perfectly if the product of their masses and their distances from the fulcrum are equal, m1 d1 = m2d2 (Figure 6.6.5)

For example, a 60 1b child 6 feet from the fulcrum will balance a 40 1b child 9 feet from the fulcrum, 60·6=40·9. If the fulcrum is at the origin x=0, the masses m1 and m2 have coordinates x1 = -d and x2 =d2. The equation for balancing becomes

m1x1 + m2 x2 =0

Similarly, finitely many masses m1,…mk at the points x1 …，xk will balance about the point x=0 if

m1x1 + …+ mkxk = 0.

Given a mass m at the point x, the quantity mx is called the moment about the origin.

The moment of a finite collection of point masses m1,…mk at x1…xk about the origin is defined as the sum

M = m1x1 + …mk xk.

Suppose the point masses are rigidly connected to a rod of mass zero. If the moment M is equal to zero, the masses will balance at the origin. In general they will balance at a point ______ called the center of gravity (Figure 6.6.6) . _____ is equal to the moment divided by the total mass m.

Since the mass m is positive, the moment M has the same sign as the center of gravity ___.

Figure 6.6.6

Now consider a length of wire between x= a and x= b whose density at x is ρ(x). The moment of the wire about the origin is defined as the integral

M = ____ xρ (x) dx.

This formula is justified by considering a piece of the wire of infinitesimal length Δx. On the piece from x to x +Δx the density remains infinitely close to ρ(x). Thus if ΔM is the moment of the piece,

ΔM ≈xΔm ≈xρ(x)Δx (compared to Δx).

The moment of an object is equal to the sum of the moments of its parts. Hence by the Infinite Sum Theorem,

M =________xρ(x)dx.

If the wire has moment M about the origin and mass m, the center of mass of the wire is defined as the point

A point of mass m located at ___ has the same moment about the origin as the whole wire, M = __ m. Physically, the wire will balance on a fulcrum placed at the center of mass.

EXAMPLE 3 A wire between x=0 and x=1 has density ρ (x) = x² (Figure 6.6.7). The moment is

The mass and center of mass are

Figure 6.6.7

4 MOMENTS, TWO DIMENSIONS

A mass m at the point (x0, y0) in the (x, y) plane will have moments Mx about the x- axis and MY about the y- axis (Figure 6.6.8). They are defined by

Mx = my0, My= mx0.

Consider a vertical length of wire of mass m and constant density which lies on the line x= x0 from y=a to y=b.

The wire has density

The infinitesimal piece of the wire from y to y +Δy shown in Figure 6.6.9 will have mass and moments

Δm=ρΔy,

ΔMx≈yΔm=yρΔy, (compared to Δy),

ΔMy≈ x0Δm=x0ρΔy, (compared to Δy),

The Infinite Sum Theorem gives the moments for the whole wire,

We next take up the case of a flat plate which occupies the region R under the curve y= f(x), f(x) ≥0, from x=a to x=b (Figure 6.6.10). Assume the density

ρ(x) depends only on the x-coordinate. A vertical slice of infinitesimal width Δx between x and x+Δx is almost a vertical length of wire between 0 and f(x) which has area ΔA and mass

Δm ≈ρ(x) ΔA≈ρ(x)f(x)Δx (compared to Δx). Putting the mass Δm into the vertical wire formulas, the moments are

ΔMy≈xΔm≈xρ(x) f(x) Δx (compared to Δx ),

ΔMx≈_____(f(x)+0) Δm ≈___ρ(x) f(x)² Δx (compared to Δx ),

Then by the Infinite Sum Theorem, the total moments are

The center of mass of a two-dimensional object is defined as the point (____) with coordinates

A single mass m at the point ____ will have the same moments as the two- dimensional body,

Mx= m___ , My= m__ . The object will balance on a pin placed at the center of mass.

If a two- dimensional object has constant density, the center of mass depends only on the region R which it occupies. The centroid of a region R is defined as the center of mass of an object of constant density which occupies R. Thus if R is the region below the continuous curve y= f(x) from x= a to x=b, then the centroid has coordinates.

where A is the area A =____ f(x) dx.

EXAMPLE 4

Find the centroid of the triangular region R bounded by the x- axis, the y-axis, and the line y=1-____ x shown in Figure 6.6.11 R is the region under the curve y= 1- ___ from x = 0 to x=2. The area of R is

The centroid is ______ where

Thus the centroid is the point _____.

Figure 6.6.11

The following principle often simplifies a problem in moments.

If an object is symmetrical about an axis, then its moment about that axis is zero and its center of mass lies on the axis.

PROOF Consider the y-axis. Suppose a plane object occupies the region under the curve y=f(y) from -a to a and its density at a point (x,y) is ρ (x) (Figure 6.6.12). The object is symmetric about the y-axis, so for all x between 0 and a,

F(-x) = f(x), ρ (-x) = ρ (x) .

Figure 6.6.12 symmetry about the y-axis

EXAMPLE 5 find the centroid of the semicircle y=______ (Figure 6.6.13). By symmetry, the centroid is on the y-axis, ___ =0. The area of the semicircle is __ π.

Then

Figure 6.6.13

5 WORK

A constant force F acting along a straight line for a distance s requires the amount of work

W = F s.

For example, the force of gravity on an object of mass m near the surface of the earth is very nearly a constant g times the mass, F= gm. Thus to lift an object of mass m a distance s against gravity requires the work W= gms. The following principle is useful in computing work done against gravity.

The amount of work done against gravity to move an object is the same as it would be if all the mass were concentrated at the center of mass. Moreover, the work against gravity depends only on the vertical change in position of the center of mass , not on the actual path of its motion.

That is, W= gms where s is the vertical change in the center of mass.

EXAMPLE 6 A semicircular plate of radius one, constant density, and mass m lies flat on the table.(a) how much work is required to stand it up with the straight edge horizontal on the table (Figure 6.6.14(a)) ? (b) how much work is required to stand it up with the straight edge vertical and one corner on the table (Figure 6.6.14(b))? From the previous exercise, we know that the

Figure 6.6.14

Center of mass is on the central radius 4/3 π from the center of the circle. Put the x-axis on the surface of the table.

(a) the center of mass is lifted a distance 4/3 π above the table. Therefore W= mg·4/(3π).

(b) The center of mass is lifted a distance 1 above the table, so W=mg.

Suppose a force F(s) varies continuously with the position s and acts on an object to move it from s = a to s =b. The work is then the definite integral of the force with respect to s,

W= _____ F(s) ds.

To justify this formula we consider an infinitesimal length Δs. On the interval from s to s + Δs the force is infinitely close to F(s), so the work ΔW done on this interval satisfies

ΔW≈ F(s)Δs (compared to Δs).

By the Infinite Sum Theorem,

Figure 6.6.15

EXAMPLE 8 The force of gravity between two particles of mass m1 and m2 is

F= gm1m2 / s²,

Where g is constant and s is the distance between the particles. Find the work required to move the particle m2 from a distance a to a distance b from m1 (Figure 6.6.16).

Figure 6.6.16

PROBLEMS FOR SECTION 6.6

In Problems 1-16 below, find (a) the mass, (b) the moments about the x-and y-axes, (c) the center of mass of the given object.

1 A wire on the x- axis, 0≤ x ≤2 with density ρ(x) = 2.

2 A wire on the x- axis, 0≤ x ≤4 with density ρ(x) = x3.

3 A wire on the y- axis, 0≤ y ≤4, whose density is twice the distance from the lower end of the wire times the square of the distance from the upper end.

4 A straight wire from the point (0,0) to the point (1,1) whose density at each point (x,x) is equal to

3x.

5 A wire of length 6 and constant density k which is bent in the shape of an L covering the

intervals [0, 2] on the x-axis and [0, 4] on the y- axis.

6 The plane object bounded by the x-axis and the curve y= 4-x², with constant density k.

7 The plane object bounded by the x-axis and the curve y=4-x², with density ρ(x)=x².

8 The plane object bounded by the lines x=0, y= x, y=4-3x, with density ρ(x)=2x.

9 The plane object between the x-axis and the curve y=x², 0≤x≤1,with density ρ(x)=1/x.

10 The object bounded by the x-axis and the curve y =x3, 0≤x≤1, with density ρ(x)=1-x².

11 The object bounded by the x-axis and the curve y=1/x, 1≤x≤2, with density ρ(x)=____.

12 The disc bounded by x²+y² =4 with density ρ(x)=____.

13 The object in the top half of the circle x²+y² =1, with density ρ(x)=2|x|.

14 The object between the x-axis and the curve y=_______, with density equal to the cube of the

distance from the y - axis.

15 The object bounded by the x-axis and the curve y=4x-x², with density ρ(x)=2x.

16 The object bounded by the curves y= - f(x) and y= f(x), 0≤x≤3, with density ρ(x)=4/f(x).(f(x) is

always positive.)

In Problems 17-24, sketch and find the centroid of the region bounded by the given curves.

17 y=0, y=2, -1≤ x≤ 5 18 y=0, x=0, 3x+4x=12

19 y=0, y=1- x² 20 y=0, y=1- x² , 0 ≤ x ≤ 1

21 y=0, y=________ 22 y=0, y=________ , 0 ≤ x≤ 3

23 y=0, y=x1/3,0 ≤ x ≤ 1

24 x=0, y=0, ____ +___ =1, first quadrant

25 Find the mass of an object in the region under the curve y=sin x, 0 ≤ x≤ π, with density

ρ(x)=cos²x.

26 Find the mass of an object in the region between the curve y=sin x cos x, y=sin x,

0 ≤ x≤ π/2, with density ρ(x)=cos x.

27 Find the mass of an object in the region under the curve y=ex, -1 ≤ x≤ 1, with density

e1-2x.

28 Find the mass of an object in the region under the curve y=1n x, 1 ≤ x≤ e, with density

ρ(x)=1/x.

29 Find the centroid of the region under the curve y=x-2, 1≤ x ≤2.

30 Find the centroid of the region under the curve y=1/_______, 1≤ x ≤4.

31 Find the centroid of the region bounded y=0, y=x(1-x²), 0≤ x ≤ 1.

□32 Show that the moments of an object bounded by the two curves y= f(x) and y= g(x),

a≤x≤b,are

33 Use the formulas in Problem 32 to find the centroid of the region between the curves y=x² and

y=x.

34 A piece of metal weighing 50 1bs is in the shape of a triangle of sides 3, 4, and 5ft. Find the

amount of work required to stand the piece up on(a) the 3 ft side, (b) the 4ft side.

35 A 4 ft chain lies flat on the ground and has constant density of 51bs/ft. How much work is

required to lift one end 6 ft above the ground?

36 In Problem 35, how much work is required to lift the center of the chain 6 ft above the ground?

37 A 4 ft chain has a density of 4x 1bs/ft at a point x ft from the left end. How much work is

needed to lift the left 6 ft above the ground?

38 In Problem 37, how much work is needed to lift both ends of the chain to the same point 6 ft

above the ground?

39 A spring exerts a force of 4x 1bs when compressed a distance x. How much work is needed to compress the spring 5 ft from its natural length?

40 A bucket of water weighs 10 1bs and is tied to a rope which has a density of ___ 1b/ft.

How much work is needed to lift the bucket from the bottom of a 20 ft well?

41 The bucket in Problem 40 is leaking water at the rate of ___ 1b/sec and is raised from the well

bottom at the rate of 4/ft sec. How much work is expended in lifting the bucket?

42 Two electrons repel each other with a force inversely proportional to the square of the distance between them, F= k/s². If one electron is held fixed at the origin, find the work required to move a second electron along the x-axis from the point (10,0) to the point (5,0).

43 If one electron is held fixed at the point (0,0) and another at the point (100,0), find the work required to move a third electron along the x-axis from (50,0) to (80,0).