# Get questions and answers for Classical Dynamics Of Particles

## GET Classical Dynamics Of Particles TEXTBOOK SOLUTIONS

1 Million+ Step-by-step solutions*Q:In Exercise 2, if the 35-N force acted downward atIn Exercise 2, if the 35-N force acted downward at an angle of 40° relative to the horizontal, what would be the acceleration in this case?*

Q:In a pole-sliding game among friends, a 90-kg man makesIn a pole-sliding game among friends, a 90-kg man makes a total vertical drop of 7.0 m while gripping the pole which exerts and upward force (call it Fp) on him. Starting from rest and sliding with a constant acceleration, his slide takes 2.5 s.

(a) Draw the man’s free body diagram being sure to label all the forces.

(b) What is the magnitude of the upward force exerted on the man by the pole?

(c) A friend whose mass is only 75 kg, slides down the same distance, but the pole force is only of the force on his buddy. How long did the second per-son’s slide take?

Q:A book is sitting on a horizontal surface. (a) ThereA book is sitting on a horizontal surface.

(a) There is (are) (1) one, (2) two, or (3) three force(s) acting on the book.

(b) Identify the reaction force to each force on the book.

Q:In an Olympic figure-skating event, a 65-kg male skater pushesIn an Olympic figure-skating event, a 65-kg male skater pushes a 45-kg female skater, causing her to accelerate at a rate of 2.0 m/s2. At what rate will the male skater accelerate? What is the direction of his acceleration?

Q:A sprinter of mass 65.0 kg starts his race byA sprinter of mass 65.0 kg starts his race by pushing horizontally backward on the starting blocks with a force of 200 N.

(a) What force causes him to accelerate out of the blocks: (1) his push on the blocks; (2) the downward force of gravity; or (3) the force the blocks exert forward on him?

(b) Determine his initial acceleration as he leaves the blocks.

Q:Jane and John, with masses of 50 kg and 60Jane and John, with masses of 50 kg and 60 kg, respectively, stand on a frictionless surface 10 m apart. John pulls on a rope that connects him to Jane, giving Jane an acceleration of 0.92 m/s2 toward him.

(a) What is John’s acceleration?

(b) If the pulling force is applied constantly, where will Jane and John meet?

Q:The displacement in meters of a certain vibrating mass isThe displacement in meters of a certain vibrating mass is described by x(t) = 0.005 sin 6t. What is the amplitude and frequency of its velocity (f)? What is the amplitude and frequency of its acceleration x(t)7

Q:The distance a spring stretches from its “free length” isThe distance a spring stretches from its “free length” is a function of how much tension is applied to it. The following table gives the spring length y that was produced in a particular spring by the given applied force f. The spring’s free length is 4.7 in. Find a functional relation between / and x, the extension from the free length (x = y – 4.7).

Force / (pounds)Spring length y (inches)

0……………………………………….4.7

0.47…………………………………….7.5

1.15……………………………………10.6

1.64……………………………………12.9

Q:The following “small angle” approximation for the sine is usedThe following “small angle” approximation for the sine is used in many engineering applications to obtain a simpler model that is easier to understand and analyze. This approximation states that sin x ( x, where x must be in radians. Investigate the accuracy of this approximation by creating three plots. For the first plot, plot sin x and x versus x for 0 Q:Obtain two linear approximations of the function f(0) = sinObtain two linear approximations of the function f(0) = sin 0, one valid near θ = π/4 rad and the other valid near θ = 3π/4 rad.

Q:Obtain two linear approximations of the function f(θ) = cosObtain two linear approximations of the function f(θ) = cos θ, one valid near θ = π/3 rad and the other valid near θ = 2π/3 rad

Q:Obtain a linear approximation of the function f(h) = √h,Obtain a linear approximation of the function f(h) = √h, valid near h = 25.

Q:Folklore has it that Sir Isaac Newton formulated the lawFolklore has it that Sir Isaac Newton formulated the law of gravitation supposedly after being hit on the head by a falling apple. The weight of an apple depends strongly on its variety, but a typical weight is 1 Newton! Calculate the total mass of 100 apples in kilograms. Then convert the total weight to pounds and the total mass to slugs.

Q:Obtain two linear approximations of the function f(r) = r2,Obtain two linear approximations of the function f(r) = r2, one valid near r = 5 and the other valid near r = 10.

Q:Obtain a linear approximation of the function f(h) = √hObtain a linear approximation of the function f(h) = √h valid near h = 16. Noting that f(h) ≥ 0, what is the value of h below which the linearized model loses its meaning?

Q:The flow rate f in m3/s of water through aThe flow rate f in m3/s of water through a particular pipe, as a function of the pressure drop p across the ends of the pipe (in N/m2) is given by f = 0.002√p. Obtain a linear model of f as a function of p that always underestimates the flow rate over the range 0 ≤ p ≤ 900.

In the following problems for Section 1.4, plot the data on suitable axes, and solve the problem by drawing a straight line by eye using a straightedge.

Q:In each of these problems, plot the data and determineIn each of these problems, plot the data and determine the best function y(x) (linear, exponential, or power function) to describe the data.

a.

Q:In a pole-sliding game among friends, a 90-kg man makesIn a pole-sliding game among friends, a 90-kg man makes a total vertical drop of 7.0 m while gripping the pole which exerts and upward force (call it Fp) on him. Starting from rest and sliding with a constant acceleration, his slide takes 2.5 s.

(a) Draw the man’s free body diagram being sure to label all the forces.

(b) What is the magnitude of the upward force exerted on the man by the pole?

(c) A friend whose mass is only 75 kg, slides down the same distance, but the pole force is only of the force on his buddy. How long did the second per-son’s slide take?

Q:A book is sitting on a horizontal surface. (a) ThereA book is sitting on a horizontal surface.

(a) There is (are) (1) one, (2) two, or (3) three force(s) acting on the book.

(b) Identify the reaction force to each force on the book.

Q:In an Olympic figure-skating event, a 65-kg male skater pushesIn an Olympic figure-skating event, a 65-kg male skater pushes a 45-kg female skater, causing her to accelerate at a rate of 2.0 m/s2. At what rate will the male skater accelerate? What is the direction of his acceleration?

Q:A sprinter of mass 65.0 kg starts his race byA sprinter of mass 65.0 kg starts his race by pushing horizontally backward on the starting blocks with a force of 200 N.

(a) What force causes him to accelerate out of the blocks: (1) his push on the blocks; (2) the downward force of gravity; or (3) the force the blocks exert forward on him?

(b) Determine his initial acceleration as he leaves the blocks.

Q:Jane and John, with masses of 50 kg and 60Jane and John, with masses of 50 kg and 60 kg, respectively, stand on a frictionless surface 10 m apart. John pulls on a rope that connects him to Jane, giving Jane an acceleration of 0.92 m/s2 toward him.

(a) What is John’s acceleration?

(b) If the pulling force is applied constantly, where will Jane and John meet?

Q:The displacement in meters of a certain vibrating mass isThe displacement in meters of a certain vibrating mass is described by x(t) = 0.005 sin 6t. What is the amplitude and frequency of its velocity (f)? What is the amplitude and frequency of its acceleration x(t)7

Q:The distance a spring stretches from its “free length” isThe distance a spring stretches from its “free length” is a function of how much tension is applied to it. The following table gives the spring length y that was produced in a particular spring by the given applied force f. The spring’s free length is 4.7 in. Find a functional relation between / and x, the extension from the free length (x = y – 4.7).

Force / (pounds)Spring length y (inches)

0……………………………………….4.7

0.47…………………………………….7.5

1.15……………………………………10.6

1.64……………………………………12.9

Q:The following “small angle” approximation for the sine is usedThe following “small angle” approximation for the sine is used in many engineering applications to obtain a simpler model that is easier to understand and analyze. This approximation states that sin x ( x, where x must be in radians. Investigate the accuracy of this approximation by creating three plots. For the first plot, plot sin x and x versus x for 0 Q:Obtain two linear approximations of the function f(0) = sinObtain two linear approximations of the function f(0) = sin 0, one valid near θ = π/4 rad and the other valid near θ = 3π/4 rad.

Q:Obtain two linear approximations of the function f(θ) = cosObtain two linear approximations of the function f(θ) = cos θ, one valid near θ = π/3 rad and the other valid near θ = 2π/3 rad

Q:Obtain a linear approximation of the function f(h) = √h,Obtain a linear approximation of the function f(h) = √h, valid near h = 25.

Q:Folklore has it that Sir Isaac Newton formulated the lawFolklore has it that Sir Isaac Newton formulated the law of gravitation supposedly after being hit on the head by a falling apple. The weight of an apple depends strongly on its variety, but a typical weight is 1 Newton! Calculate the total mass of 100 apples in kilograms. Then convert the total weight to pounds and the total mass to slugs.

Q:Obtain two linear approximations of the function f(r) = r2,Obtain two linear approximations of the function f(r) = r2, one valid near r = 5 and the other valid near r = 10.

Q:Obtain a linear approximation of the function f(h) = √hObtain a linear approximation of the function f(h) = √h valid near h = 16. Noting that f(h) ≥ 0, what is the value of h below which the linearized model loses its meaning?

Q:The flow rate f in m3/s of water through aThe flow rate f in m3/s of water through a particular pipe, as a function of the pressure drop p across the ends of the pipe (in N/m2) is given by f = 0.002√p. Obtain a linear model of f as a function of p that always underestimates the flow rate over the range 0 ≤ p ≤ 900.

In the following problems for Section 1.4, plot the data on suitable axes, and solve the problem by drawing a straight line by eye using a straightedge.

Q:In each of these problems, plot the data and determineIn each of these problems, plot the data and determine the best function y(x) (linear, exponential, or power function) to describe the data.

a.

b.

c.

Plot the data and obtain a function that describes the data. Estimate when the population will be double its 2005 size.

C(t)/C(0) = e-bt

a. Use the half-life of carbon-14 to find the value of the parameter b and plot the function.

b. Suppose we estimate that 90% of the original carbon-14 remains. Estimate how long ago the organism died.

c. Suppose our estimate of h is off by ± 1 %. How does this affect the age estimate in part (b)?

Q:Quenching is the process of immersing a hot metal objectQuenching is the process of immersing a hot metal object in a bath for a specified time to improve properties such as hardness. A copper sphere 25 mm in diameter, initially at 300Â°C, is immersed in a bath at 0Â°C. Measurements of the sphere’s temperature versus time are shown here. Plot the data and find a functional description of the data.

Q:The useful life of a machine bearing depends on itsThe useful life of a machine bearing depends on its operating temperature, as shown by the following data. Plot the data and obtain a functional description of the data. Estimate a bearing’s life if it operates at 150Â°F.

Q:A certain electric circuit has a resistor and a capacitor.A certain electric circuit has a resistor and a capacitor. The capacitor is initially charged to 100 V. When the power supply is detached, the capacitor voltage decays with time as shown in the following data table. Find a functional description of the capacitor voltage v as a function of time

Q:Water (of volume 425 ml) in a glass measuring cupWater (of volume 425 ml) in a glass measuring cup was allowed to cool after being heated to 207 F. The ambient air temperature was 70Â°F. The measured water temperature at various times is given in the following table.

Obtain a functional description of the relative water temperature ((T = T – 70) versus time.

Obtain a functional description of the volume outflow rate / through the straw as a function of water height h above the hole.

Obtain a functional description of the volume outflow rate f through the straw as a function of water height h above the hole.

X(s) = (7s + 4)/(2s2 + 16s + 30)

Q:Solve each of the following problems with the trial solutionSolve each of the following problems with the trial solution method. Identify the free, forced, transient, and steady-state responses.

a. + 8 + 15A = 30

x(0) = 10

(0) = 4

b. + 10 + 25A = 75

x(0) = 10

(0) = 4

c. + 25 =100

x(0) = 10

(0) = 4

d. + 8 + 65A = 130

x(0) = 10

(0) = 4

Q:Determine whether the following models are stable, unstable, or neutrallyDetermine whether the following models are stable, unstable, or neutrally stable:

a. 3 -5x = 12

b. – 3 – 10x = 50

c. – 6 + 34x = 68

d. = 3

e. + 4x = 5

f. + 5 = 7

Q:(a) Prove that the second-order system whose characteristic polynomial is(a) Prove that the second-order system whose characteristic polynomial is ms2 + cs + k is stable if and only if m, c, and k have the same sign,

(b) Derive the conditions for neutral stability.

Q:Obtain the steady-state response of each of the following models,Obtain the steady-state response of each of the following models, and estimate how long it will take the response to reach steady state.

a. 13 + 4x = 16us(t)……………. x(0) = 0

b. I3+4x = 16us(t)……………… x(0) = 1

c. 15 – 7x = 14us,(t)……………..x(0) = -2

Q:Compare the responses of 4 + x = (t)Compare the responses of 4 + x = (t) + g(t) and 4 + x = g(t) if g(t) = 5us,(t) and x(0-) = 0.

Q:Solve each of the following problems by separation of variables: a.Solve each of the following problems by separation of variables:

a. + 5×2 = 25……………… x(0) = 3

b. -4×2 = 36…………………x(0)=l0

c. x – 5x = 25………………. x(0) = 4

d. + 2e-4tx = 0………………x(0) = 5

Q:If applicable, compute (, τ, ωn, and ωd for theIf applicable, compute (, τ, ωn, and ωd for the following characteristic polynomials. If not applicable, state the reason why.

a. s2 + 4s + 40 = 0

b. s2 – 2s + 24 = 0

c. s2+ 20s + 100 = 0

d. s +10 = 0

Q:The characteristic equation of a certain system is s2 +The characteristic equation of a certain system is s2 + 10ds + 29d2 = 0, where d is a constant,

(a) For what values of d is the system stable?

(b) Is there a value of d for which the free response will consist of decaying oscillations?

Q:For each of the following equations, determine the transfer functionFor each of the following equations, determine the transfer function X(s)/F(s) and compute the characteristic roots:

a. 5 + 7x = 15 f(t)

b. 3 + 30 + 63x = 5f(t)

c. + 10 + 21x =4 (t)

d. + 14 + 49A = 7 (t)

e. + 14 + 58x = 6 (t) + 4f (t)

f. 5x+ 7x = 4 (t) + 15 f(t)

Q:Obtain the transfer functions X(s)/F(s) and Y(s)/F(s) for the followingObtain the transfer functions X(s)/F(s) and Y(s)/F(s) for the following model:

3 = y

= f(t) -3y- 15x

Q:Obtain the transfer functions X(s)/F(s) and Y(s)/F(s) for the followingObtain the transfer functions X(s)/F(s) and Y(s)/F(s) for the following model:

= -2x + 5y

= f(t) – 6y – 4x

Q:A. Obtain the transfer functions X(s)/F(s) and Y(s)/F(s) for thea. Obtain the transfer functions X(s)/F(s) and Y(s)/F(s) for the following model:

4 = y

= f(t) – 3y – 12x

b. Compute τ, (, ωn, and ωd for the model.

c. If f(t) = us(t), will the responses x(t) and y(t) oscillate? If so, compute the radian oscillation frequency, and estimate how long it will take for the oscillations to disappear.

d. Suppose that f(t) = us(t) and x(0) = (0) = 0. Obtain x(t) and y(t).

Q:A. Obtain the transfer functions X(s)/F(s) and X(s)/G(s) for thea. Obtain the transfer functions X(s)/F(s) and X(s)/G(s) for the following model:

= -4x+ 2y + f(t)

=-9y – 5x + g(t)

b. Compute τ, (, ωn, and ωd for the model.

c. If f(t) = g(t) = 0, estimate how long it will take for the responses x(t) and y(t) to disappear.

Q:Solve the following problems for x(t). Compare the values ofSolve the following problems for x(t). Compare the values of x(0+) and A(0-). For parts (b) through (d), also compare the values of i(0+) and i(0-).

a. 7 + 5x = 4δ(t)

x(0-) = 3

b. 3 + 30 + 63x = 5δ(t)

x(0-) = (0-) = 0

c. + 14 + 49x = 3δ(t)

x(0-) = 2

(0-) = 3

d. + 14 + 58x = 4δ(t)

x(0-) = 4

(0-) = 7

Q:Solve the following problems for x(t). The input g(t) isSolve the following problems for x(t). The input g(t) is a unit-step function, g(t) = us(t). Compare the values of x(0+) and x(0-). For parts (c) and (d), also compare the values of (0+) and i(0-).

a. 7 + 5x = 4(t)

x(0-) = 3,

g(0-) = 0

b. 7 + 5x = 4(t) + 6g (t)

x(0-) = 3,

g(0-) = 0

c. 3 + 30 + 63x = 4(t)

x(0-) = 2

(0-) = 3,

g(0-) = 0

d. 3 + 30 + 63x =4(t) + 6g(t)

x(0-) = 4

(0-) = 7,

g(0-) = 0

Q:Solve the following problem for x(t) and y(t): 3 = y x(0)Solve the following problem for x(t) and y(t):

3 = y

x(0) = 5

= 4us,(t) – 3y – I5x

u(0)=l0

Q:Derive the Laplace transform of the ramp function x(t) =Derive the Laplace transform of the ramp function x(t) = mt, whose slope is the constant m.

Q:Solve the following problem for x(t) and y(t): = -2xSolve the following problem for x(t) and y(t):

= -2x + 5y

x(0) = 5

= -6y – 4x + 10u(t)

y(0) = 2

Q:Determine the general form of the solution of the followingDetermine the general form of the solution of the following equation, where the initial conditions y(0) and (0) have arbitrary values:

+ y = e-t

Q:A. Use the Laplace transform to obtain the form ofa. Use the Laplace transform to obtain the form of the solution of the following equation:

+ 4x = 3t

b. Obtain the solution to the equation in part (a) for the following conditions:

x(0) = 10, x(5) = 30.

Q:Obtain the inverse Laplace transform of X(s) = 30/((s2+6s+34)(s2+36))Obtain the inverse Laplace transform of

X(s) = 30/((s2+6s+34)(s2+36))

Q:Solve the following problem for x(t). + 12 + 40xSolve the following problem for x(t).

+ 12 + 40x = 3 sin 5t

x(0) = (0) = 0

Q:Obtain the inverse transform in the form x(t) = AObtain the inverse transform in the form x(t) = A sin(ωt +(), where A > 0.

X(s) = (4s + 9)/(s2 + 25)

Q:Use the Laplace transform to solve the following problem: +Use the Laplace transform to solve the following problem:

+ 6+ 34x = 5 sin 6t

x(0) = 0

(0) = 0

Q:Express the oscillatory part of the solution of the followingExpress the oscillatory part of the solution of the following problem in the form of a sine function with a phase angle:

+ 12 + 40x = 10

x(0) = (0) = 0

Q:Find the steady-state difference between the input f(t) and theFind the steady-state difference between the input f(t) and the response x(t), if f(t) = 6t.

+8+ x = f(t)

x(0) = (0) = 0

Q:Invert the following transform: X(s) = (1 -e-3s)/(s2 + 6s +8)Invert the following transform:

X(s) = (1 -e-3s)/(s2 + 6s +8)

Q:Extend the results of Problem 2.4 to obtain the LaplaceExtend the results of Problem 2.4 to obtain the Laplace transform of t2.

Q:Obtain the Laplace transform of the function plotted in Figure.Obtain the Laplace transform of the function plotted in Figure.

Q:Obtain the Laplace transform of the function plotted in FigureObtain the Laplace transform of the function plotted in Figure

Q:Obtain the Laplace transform of the function plotted in Figure.Obtain the Laplace transform of the function plotted in Figure.

Q:Obtain the response x(t) of the following model, where theObtain the response x(t) of the following model, where the input P(t) is a rectangular pulse of height 3 and duration 5:

4+ x = P(t)

x(0) = 0

Q:The Taylor series expansion tan t = t + t3/3 +The Taylor series expansion

tan t = t + t3/3 + 2t5/15 + 17t7/315 + …… |t| Q:Derive the initial value theorem: Lims→( sX(s) = x(0+)Derive the initial value theorem:

Lims→( sX(s) = x(0+)

Q:Derive the final value theorem: Lims→0 sX(s) = Limt→( x(t)Derive the final value theorem:

Lims→0 sX(s) = Limt→( x(t)

Q:Derive the integral property of the Laplace transform:Derive the integral property of the Laplace transform:

Q:Use MATLAB to obtain the inverse transform of the following.Use MATLAB to obtain the inverse transform of the following. If the denominator of the transform has complex roots, express x(t) in terms of a sine and a cosine.

a. X(s) = (8s + 5)/(2s2 + 20s + 48)

b. X(s) = (4s + 13)/(s2+ 8s + 116)

c. X(s) = (3s + 2)/(s2(5 + 10))

d. X(s) = (s3 + s + 6)/(s4(s + 2))

e. X(s) = (4s + 3)/(s(s2 + 6s + 34))

f. X(s) = (5s2 + 3s + 7)/(s3 + I2s2 +44s + 48)

Q:Use MATLAB to obtain the inverse transform of the following.Use MATLAB to obtain the inverse transform of the following. If the denominator of the transform has complex roots, express x(t) in terms of a sine and a cosine.

a. X(s) = 5/((s + 4)2(s + l))

b. X(s) = (4s + 9)/((s2 + 6s + 34)(s2+4s + 20))

Q:Obtain the Laplace transform of the following functions: a. x (t)Obtain the Laplace transform of the following functions:

a. x (t) = 10 + t2

b. x(t) = 6te-5t + e-3t

c. x(t) = te-3t sin 5t

d.

Q:Use MATLAB to solve for and plot the unit-step responseUse MATLAB to solve for and plot the unit-step response of the following models:

a. 3+ 21+ 30x = f(t)

b. 5 + 20+ 65x = f(t)

c. A + 32 + 60x = 3 (t) + 2f(t)

Q:Use MATLAB to solve for and plot the response ofUse MATLAB to solve for and plot the response of the following models for 0 ≤ t ≤ 1.5, where the input is f(t) = 5t and the initial conditions are zero:

a. 3+ 21+ 30x = f(t)

b. 5+ 20+ 65x = f(t)

c. 4 + 32 + 60x = 3 (t) + 2f(t)

Q:Use MATLAB to solve for and plot the response ofUse MATLAB to solve for and plot the response of the following models for 0 ≤ / ≤ 6, where the input is f(t) = 6 cos 3t and the initial conditions are zero:

a. 3+ 21+ 30x = f(t)

b. 5 + 20 + 65x = f(t)

c. 4 + 32 + 60x = 3 (t) + 2f(t)

Q:Obtain the Laplace transform of the function shown in Figure.Obtain the Laplace transform of the function shown in Figure.

Q:Obtain the inverse Laplace transform f(t) for the following: a. 6/(s2+9) b.Obtain the inverse Laplace transform f(t) for the following:

a. 6/(s2+9)

b. 5/(x2+4) + 4s/(s2+4)

c. 6/(x2+4s+13)

d. 5/(s(s+3))

e. 10/((s+3)(s+7))

f. (2s+8)/((s+3)(s+7))

Q:Obtain the inverse Laplace transform f(t) for the following: a. 5s/(s2+9) b.Obtain the inverse Laplace transform f(t) for the following:

a. 5s/(s2+9)

b. 6/(s2-9)

c. 45/(s(s+3)2)

d. 2/(s(s2+4s+13)

e. 20/(s(s2+4)

f. 20s/(s2+4)

Q:Determine whether or not the following equations are linear orDetermine whether or not the following equations are linear or nonlinear, and state the reason for your answer.

a. y + 5 + y = 0

b. + sin y = 0

c + √y = 0

d. + 5t2 + 3y = 0

e. + 3t2 sin y = 0

f. + ety = 0

Q:Use the initial and final value theorems to determine x(0+)Use the initial and final value theorems to determine x(0+) and x(() for the following transforms:

a. X(s) = 5/(3s+7)

b. X(s) = 10/(3s2+7s+4

Q:Obtain the inverse Laplace transform x(t) for each of theObtain the inverse Laplace transform x(t) for each of the following transforms:

a. X(s) = 6/(s(s+4)

b. X(s) = (125 + 5)/(s(s+3)

c. X(s) = (4s+7)/(s+2)(s+5))

d. X(s) = 5/(s2(2s+8))

e. X(s) = (3s+2)/(s2(s+5))

f. X(s) = (12s+5) /((s+3)2(s+7))

Q:Obtain the inverse Laplace transform x(t) for each of theObtain the inverse Laplace transform x(t) for each of the following transforms:

a. X(s) = (7s + 2)/(s2 + 6s + 34)

b. (4s + 3)/(s(s2 + 6s + 34)

c. X(s) = (4s+9)/((s2+6s+34)(s2+4s+20))

d. X(s) = (5s2 + 3s + 7)/(s3 + 12s2 + 44s+48)

Q:Solve the following problems: a. 5 = 7t x(0) = 3 b. 4Solve the following problems:

a. 5 = 7t

x(0) = 3

b. 4 = 3e-5t

x(0) = 4

c. 7:=4t

x(0) = 3

(0) = 5

d. 3 = 8e-4t

x(0) =3

(0) = 5

Q:Solve the following problems: a. 5 + 7x = 0 ……………………….Solve the following problems:

a. 5 + 7x = 0 ………………………. x(0) = 4

b. 5 + 7x = 15……………………… x(0) = 0

c. 5 + 7x = 15……………………… x(0) =4

d. + 7x =4t ……………………….. x(0) = 5

Q:Solve the following problems: a. + 10 + 21x =0 x(0)Solve the following problems:

a. + 10 + 21x =0

x(0) = 4

(0) = -3

b. + 14 + 49x = 0

x(0) = 1

(0) = 3

c. + 14 + 58x = 0

x(0) = 4

(0) =-8

Q:Solve the following problems: a. + 7 + 10x =Solve the following problems:

a. + 7 + 10x = 20

x(0) = 5

(0) = 3

b. 5 + 20 + 20x = 28

x(0) = 5

(0) = 8

c. + 16x = 144

x(0) = 5

(0) = 12

d. + 6 + 34x = 68

x(0) = 5

(0) = 7

Q:Solve the following problems: a. 3 + 30 + 63x =Solve the following problems:

a. 3 + 30 + 63x = 5

x(0) = (0) = 0

b. + 14+ 49x = 98

x(0) = (0) = 0

c. + 14 + 58x = 174

x(0) = (0) = 0

Q:Solve the following problems where x(0) = (0) = 0. a.Solve the following problems where x(0) = (0) = 0.

a. + 8 + 12x = 60

b. + 12 + 144x = 288

c. + 49x=147

d. + 14 + 85x = 170

Q:Invert the following transforms: a. 6/(s(s+5) b. 4/(s + 3)(s + 8) c.Invert the following transforms:

a. 6/(s(s+5)

b. 4/(s + 3)(s + 8)

c. (8s + 5)/(2s2 + 20s+48)

d. 4s + 13/(s2 + 8s+ 116)

Q:Solve each of the following problems by direct integration: a.Solve each of the following problems by direct integration:

a. 4 = 3t

x(0) = 2

b. 5x = 2e-4t

x(0) = 3

c. 3 = 5t

x(0) = 2

(0) = 7

d. 4 = 7e-2t

x(0) = 4

(0) = 2

e. = 0

x(0) = 2

(0) = 5

Q:Invert the following transforms: a. (3s + 2)/(s2(s+10)) b. 5/((s+4)2(s + l)) c.Invert the following transforms:

a. (3s + 2)/(s2(s+10))

b. 5/((s+4)2(s + l))

c. (s2 + 3s + 5)/(s3(s + 2))

d. (s3 + s + 6)/(s4(s + 2))

Q:Solve the following problems for x(t): a. 5 + 3x =Solve the following problems for x(t):

a. 5 + 3x = 10 + t2

x(0) = 2

b. 4 + 7x = 6te-5t + e-3t

x(0) = 5

c. 4 + 3x = te-3t sin 5t

x(0) = 10

(0) = -2

Q:Consider the falling mass in Example 3.1.1 and Figure 3.1.2.Consider the falling mass in Example 3.1.1 and Figure 3.1.2. Find its speed and height as functions of time. How long will it take to reach (a) the platform and (b) the ground?

Q:The two masses shown in Figure are released from rest.The two masses shown in Figure are released from rest. The mass of block A is 100 kg; the mass of block B is 20 kg. Ignore the masses of the

pulleys and rope. Determine the acceleration of block A and of block B. Do they rise or fall?

Q:Instead of using the system shown in Figure 3.2.6a toInstead of using the system shown in Figure 3.2.6a to raise the mass m2, an engineer proposes to use two simple machines, the pulley and the inclined plane, to reduce the weight required to lift m2. The proposed design is shown in Figure. The pulley inertias are negligible. The available horizontal space limits the angle of the inclined plane to no less than 30Â°.

a. Suppose that the friction between the plane and the mass W2 is negligible. Determine the smallest value m1 can have to lift m2. Your answer should be a function of m2 and Î¸.

b. In practice, the coefficient of dynamic friction Î¼d between the plane and the mass m2 is not known precisely. Assume that the system can be started to overcome static friction. For the value of m1 = m2/2, how large can Î¼d be before m1 cannot lift m2?

Q:An inextensible cable with a tension force f = 500An inextensible cable with a tension force f = 500 N is used to pull a two-wheeled cart on a horizontal surface. The wheels roll without slipping. The cart has a mass mc = 120 kg. Each wheel has a radius RW = 0.4 m and an inertia Iw. = 1.5 kg.m2. Disregard the mass of the axle. The center of mass of the system is at point G. Compute the translational acceleration v of the cart.

Q:Consider the cart shown in Figure. Suppose we model theConsider the cart shown in Figure. Suppose we model the wheels as solid disks. Then the wheel inertia is given by IW = mWRw2/2. How small must the wheel mass mw be relative to the cart body mass mc so that the two wheels increase the equivalent mass me by no more than 10%?

Q:Consider the spur gears shown in Figure, where I1 =Consider the spur gears shown in Figure, where I1 = 0.3 kg.m2 and I2 = 0.5 kg.m2. Shaft 1 rotates three times faster than shaft 2. The torques are given as T1 = 0.5 N.m and T2 = -0.3 N.m Compute the equivalent inertia on shaft 2, and find the angular acceleration Ï2-

Q:Consider the spur gears shown in Figure, where I1 =Consider the spur gears shown in Figure, where I1 = 0.1 kg-m2 and I2 = 2 kg-m2. Shaft 1 rotates twice as fast as shaft 2. The torques are given as T1 = 10 N-m and T2 = 0.

(a) Compute the angular acceleration Ï1 assuming the shaft and gear inertias are negligible,

(b) Suppose the gear on shaft 1 has inertia 0.005 kg-m2 and the gear on shaft 2 has inertia 0.05 kg-m2. Compute the angular acceleration Ï1.

Q:Derive the expression for the equivalent inertia I1, felt onDerive the expression for the equivalent inertia I1, felt on the input shaft, for the spur gears shown in Figure, where the combined gear-shaft inertias are Is1 and Is2.

Q:Draw the free body diagrams of the two spur gearsDraw the free body diagrams of the two spur gears shown in Figure. Use the resulting equations of motion to show that T2 = NT1 if the gear inertias are negligible or if there is zero acceleration. Here T2 is taken to be the torque felt on shaft 2 due to the applied torque T1.

Q:The geared system shown in Figure represents an elevator system.The geared system shown in Figure represents an

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