Schaum’s Outlines of Differential Equations 3rd. Edition by Bronson, Richard, Costa, Gabriel B | 9780071491105 | 0071456872

Schaum’s Outlines of Differential Equations [EXP-71344]

552 SOLVED PROBLEMS

Question: 14.9

A 10-kg mass is attached to a spring having a spring constant of 140 N/m. The mass is started in motion from the equilibrium position with an initial velocity of 1 m/sec in the upward direction and with an applied external force F(t) = 5 sin t. Find the subsequent motion of the mass if the force

Verified Answer:

Here

m=10, k=140, a=90

, and F(t) = ...

Question: 14.5

A 10-kg mass is attached to a spring, stretching it 0.7 m from its natural length. The mass is started in motion from the equilibrium position with an initial velocity of 1 m/sec in the upward direction. Find the subsequent motion, if the force due to air resistance is -90x N.

Verified Answer:

Taking

g=9.8  m / sec ^{2}

, we have...

Question: 14.10

A 128-lb weight is attached to a spring having a spring constant of 64 lb/ft. The weight is started in motion with no initial velocity by displacing it 6 in above the equilibrium position and by simultaneously applying to the weight an external force F(t) = 8 sin 4t. Assuming no air resistance,

Verified Answer:

Here

m=4, k=64, a=0, \text { and } F(t)=8 \...

Question: 7.18

A 50-gal tank initially contains 10 gal of fresh water. At t = 0, a brine solution containing 1 Ib of salt per gallon is poured into the tank at the rate of 4 gal/min, while the well-stirred mixture leaves the tank at the rate of 2 gal/min. Find (a) the amount of time required for overflow to occur

Verified Answer:

(a) Here

a=0, b=1, e=4, f=2 \text {, and } ...

Question: 7.4

A bacteria culture is known to grow at a rate proportional to the amount present. After one hour, 1000 strands of the bacteria are observed in the culture; and after four hours, 3000 strands. Find (a) an expression for the approximate number of strands of the bacteria present in the culture at any

Verified Answer:

(a) Let N(t) denote the number of bacteria strands...

Question: 7.9

A body at a temperature of 50° F is placed outdoors where the temperature is 100° F. If after 5 minutes the temperature of the body is 60° F, find (a) how long it will take the body to reach a temperature of 75° F and (b) the temperature of the body after 20 minutes.

Verified Answer:

Using (7.2) with

T_{m}=100

(the sur...

Question: 7.10

A body at an unknown temperature is placed in a room which is held at a constant temperature of 30° F. If after 10 minutes the temperature of the body is 0° F and after 20 minutes the temperature of the body is 15° F, find the unknown initial temperature.

Verified Answer:

From (7.2),

\frac{d T}{d t}+k T=30 k[/latex...

Question: 7.15

A body of mass 2 slugs is dropped with no initial velocity and encounters an air resistance that is proportional to the square of its velocity. Find an expression for the velocity of the body at any time t.

Verified Answer:

The force due to air resistance is

-k v^{2}...

Question: 7.11

A body of mass 5 slugs is dropped from a height of 100 ft with zero velocity. Assuming no air resistance, find (a) an expression for the velocity of the body at any time t, (b) an expression for the position of the body at any time t, and (c) the time required to reach the ground.

Verified Answer:

Choose the coordinate system as in Fig. 7-5. Then,...

Question: 7.14

A body of mass m is thrown vertically into the air with an initial velocity v0. If the body encounters an air resistance proportional to its velocity, find (a) the equation of motion in the coordinate system of Fig. 7-6, (b) an expression for the velocity of the body at any time t, and (c) the time

Verified Answer:

(a) In this coordinate system, Eq. (7.4) may not b...