Questions and Answers of Classical Dynamics Of Particles

In 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?
In 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
A 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.
In 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
A 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
Jane 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
The 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
The 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 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
Obtain 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.
Obtain 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
Obtain a linear approximation of the function f(h) = √h, valid near h = 25.
Folklore 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
Obtain two linear approximations of the function f(r) = r2, one valid near r = 5 and the other valid near r = 10.
Obtain 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?
The 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
In 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.
The population data for a certain country are given here.Plot the data and obtain a function that describes the data. Estimate when the population will be double its 2005 size.
The half-life of a radioactive substance is the time it takes to decay by half. The half-life of carbon-14, which is used for dating previously living things, is 5500 years. When an organism dies, it
Quenching 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 300°C, is immersed in
The 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
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
Water (of volume 425 ml) in a glass measuring cup was allowed to cool after being heated to 207 F. The ambient air temperature was 70°F. The measured water temperature at various times is given in
Consider the milk container of Example 1.4.2 (Figure 1.4.7). A straw 19 cm long was inserted in the side of the container. While adjusting the tap flow to keep the water height constant, the time for
Consider the milk container of Example 1.4.2 (Figure 1.4.7). A straw 9.5 cm long was inserted in the side of the container. While adjusting the tap flow to keep the water height constant, the time
Compare the LCD method with equation (2.4.4) for obtaining the inverse Laplace transform ofX(s) = (7s + 4)/(2s2 + 16s + 30)
Solve each of the following problems with the trial solution method. Identify the free, forced, transient, and steady-state responses.
Determine whether the following models are stable, unstable, or neutrally stable:
(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.
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 =
Compare the responses of 4 + x = (t) + g(t) and 4 + x = g(t) if g(t) = 5us,(t) and x(0-) = 0.
Solve each of the following problems by separation of variables:a. ẋ + 5x2 = 25……………… x(0) = 3b. ẋ-4x2 = 36…………………x(0)=l0c. x ẋ - 5x = 25………………. x(0) = 4d.
If 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
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
For each of the following equations, determine the transfer function X(s)/F(s) and compute the characteristic roots:a. 5ẋ + 7x = 15 f(t)b. 3 ẍ + 30 + 63x = 5f(t)c. ẍ + 10 + 21x
Obtain the transfer functions X(s)/F(s) and Y(s)/F(s) for the following model:3ẋ = yẏ = f(t) -3y- 15x
Obtain the transfer functions X(s)/F(s) and Y(s)/F(s) for the following model: = -2x + 5y = f(t) - 6y - 4x
a. Obtain the transfer functions X(s)/F(s) and Y(s)/F(s) for the following model:4 = y = f(t) - 3y - 12xb. Compute τ, (, ωn, and ωd for the model.c. If f(t) = us(t), will the responses x(t) and
a. 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,
Solve 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-).
Solve 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-).
Solve the following problem for x(t) and y(t):
Derive the Laplace transform of the ramp function x(t) = mt, whose slope is the constant m.
Solve the following problem for x(t) and y(t):
Determine the general form of the solution of the following equation, where the initial conditions y(0) and ẏ (0) have arbitrary values:ÿ + y = e-t
a. Use the Laplace transform to obtain the form of the solution of the following equation:ẍ + 4x = 3tb. Obtain the solution to the equation in part (a) for the following conditions:x(0) = 10, x(5)
Obtain the inverse Laplace transform ofX(s) = 30/((s2+6s+34)(s2+36))
Solve the following problem for x(t).
Obtain the inverse transform in the form x(t) = A sin(ωt +(), where A > 0. X(s) = (4s + 9)/(s2 + 25)
Use the Laplace transform to solve the following problem:
Express the oscillatory part of the solution of the following problem in the form of a sine function with a phase angle:
Find the steady-state difference between the input f(t) and the response x(t), if f(t) = 6t.
Invert the following transform: X(s) = (1 -e-3s)/(s2 + 6s +8)
Extend the results of Problem 2.4 to obtain the Laplace transform of t2.
Obtain the Laplace transform of the function plotted in Figure.
Obtain the Laplace transform of the function plotted in Figure
Obtain the Laplace transform of the function plotted in Figure.
Obtain 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
The Taylor series expansiontan t = t + t3/3 + 2t5/15 + 17t7/315 + ...... |t| < π
Derive the initial value theorem: Lims→( sX(s) = x(0+)
Derive the final value theorem: Lims→0 sX(s) = Limt→( x(t)
Derive the integral property of the Laplace transform:
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.
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)
Obtain the Laplace transform of the following functions:a. x (t) = 10 + t2b. x(t) = 6te-5t + e-3tc. x(t) = te-3t sin 5td.
Use MATLAB to solve for and plot the unit-step response of the following models:
Use 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:
Use 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:
Obtain the Laplace transform of the function shown in Figure.
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))
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)
Determine whether or not the following equations are linear or nonlinear, and state the reason for your answer.
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
Obtain 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) =
Obtain 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 +
Solve the following problems:
Solve the following problems:
Solve the following problems:
Solve the following problems:
Solve the following problems:
Solve the following problems where x(0) = (0) = 0.
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)
Solve each of the following problems by direct integration:
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))
Solve the following problems for x(t):
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?
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 thepulleys and rope. Determine the acceleration of block A and
The motor in Figure lifts the mass mL by winding up the cable with a force FA. The center of pulley B is fixed. The pulley inertias are IB and Ic. Pulley C has a mass mC. Derive the equation of
Instead 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.
An 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
Consider 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
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
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
Derive 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.
Draw 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.
The geared system shown in Figure represents an elevator system. The motor has inertia I and supplies a torque T1. Neglect the inertias of the gears, and assume that the cable does not slip on the
A baseball is thrown horizontally from the pitcher's mound with a speed of 90 mph. Neglect air resistance, and determines how far the ball will drop by the time it crosses home plate 60 ft away.
Derive the expression for the equivalent inertia Ie felt on the input shaft, for the rack and pinion shown in Figure, where the shaft inertia is Is.
Derive the expression for the equivalent inertia Ie felt on the input shaft, for the belt drive treated in Example 3.3.5, where the shaft inertias connected to the sprockets are Is1 and Is2,.
For the geared system shown in Figure, proper selection of the gear ratio N can maximize the load acceleration ω2 for a given motor and load. Note that the gear ratio is defined such that ω1 =

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