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Math Advanced Math Fundamentals of Differential Equations (9th Edition)

Whether the mass returns to its equilibrium position. The mass never _ returns to its equilibrium position. Formula used: The governing equation of a mass-spring system is: F e x t = [ i n e r t i a ] y ″ + [ d a m p i n g ] y ′ + [ s t i f f n e s s ] y = m y ″ + b y ′ + k y In the case of overdamped, b 2 − 4 m k > 0 and there will be two distinct real roots to the associated auxiliary equation. For the roots r 1 and r 2 , the general solution that determines the motion will be y ( t ) = C 1 e r 1 t + C 2 e r 2 t . Calculation: Given that, mass 20 kg is attached to a spring with stiffness 200 N/m and the damping constant is 140 N-sec/m . That is, m = 20 , k = 200 and b = 140 . Thus, the governing equation of a mass-spring system F e x t = m y ″ + b y ′ + k y for 0 external force becomes, 20 y ″ + 140 y ′ + 200 y = 0 . Dividing by 20 on both sides gives, y ″ + 7 y ′ + 10 = 0 . The associated auxiliary equation is, r 2 + 7 r + 10 = 0 . Simplify r 2 + 7 r + 10 = 0 as, r 2 + 2 r + 5 r + 10 = 0 r ( r + 2 ) + 5 ( r + 2 ) = 0 ( r + 2 ) ( r + 5 ) = 0 r = − 2 , − 5 That is, the roots are r 1 = − 2 and r 2 = − 5 . Thus, the solution form y ( t ) = C 1 e r 1 t + C 2 e r 2 t becomes y ( t ) = C 1 e − 2 t + C 2 e − 5 t . Given that, the mass is displaced 25 cm to the right of the equilibrium point and a velocity of 1m/sec to the left. Convert 25cm to meters as 1 4 m . That is, y ( 0 ) = 1 4 and y ′ ( 0 ) = − 1 . Note that, the negative sign indicates to the left direction. Apply the initial conditions to obtain the value of the constants as follows: Substitute t = 0 and y ( 0 ) = 1 4 in y ( t ) = C 1 e − 2 t + C 2 e − 5 t , y ( 0 ) = C 1 e − 2 × 0 + C 2 e − 5 × 0 1 4 = C 1 ( 1 ) + C 2 ( 1 ) C 1 + C 2 = 1 4 C 1 = 1 4 − C 2 Differentiate y ( t ) = C 1 e − 2 t + C 2 e − 5 t with respect to t as, y ′ ( t ) = − 2 C 1 e − 2 t − 5 C 2 e − 5 t . Substitute t = 0 and y ′ ( 0 ) = − 1 in y ′ ( t ) = − 2 C 1 e − 2 t − 5 C 2 e − 5 t , y ′ ( 0 ) = − 2 C 1 e − 2 × 0 − 5 C 2 e − 5 × 0 − 1 = − 2 C 1 − 5 C 2 − 1 = − 2 ( 1 4 − C 2 ) − 5 C 2 ( ∵ C 1 = 1 4 − C 2 ) − 1 = − 1 2 + 2 C 2 − 5 C 2 Simplify further as, − 1 = − 1 2 − 3 C 2 3 C 2 = 1 2 C 2 = 1 6 Thus, C 1 = 1 4 − C 2 becomes, C 1 = 1 4 − 1 6 = 6 − 4 24 = 2 24 = 1 12 That is, C 1 = 1 12 and C 2 = 1 6 . Thus, the solution y ( t ) = C 1 e − 2 t + C 2 e − 5 t becomes, y ( t ) = 1 12 e − 2 t + 1 6 e − 5 t . The aim is to find the time at which the mass first return to its equilibrium position. Note that, the equilibrium position corresponds to y = 0 . That is, solve y ( t ) = 1 12 e − 2 t + 1 6 e − 5 t for y ( t ) = 0 . Note that, the value of the exponential is never zero. But, it tends to zero as t → ∞ . Thus, it is impossible for the object to reach the equilibrium position. Therefore, the mass never _ returns to its equilibrium position.

Whether the mass returns to its equilibrium position. The mass never _ returns to its equilibrium position. Formula used: The governing equation of a mass-spring system is: F e x t = [ i n e r t i a ] y ″ + [ d a m p i n g ] y ′ + [ s t i f f n e s s ] y = m y ″ + b y ′ + k y In the case of overdamped, b 2 − 4 m k > 0 and there will be two distinct real roots to the associated auxiliary equation. For the roots r 1 and r 2 , the general solution that determines the motion will be y ( t ) = C 1 e r 1 t + C 2 e r 2 t . Calculation: Given that, mass 20 kg is attached to a spring with stiffness 200 N/m and the damping constant is 140 N-sec/m . That is, m = 20 , k = 200 and b = 140 . Thus, the governing equation of a mass-spring system F e x t = m y ″ + b y ′ + k y for 0 external force becomes, 20 y ″ + 140 y ′ + 200 y = 0 . Dividing by 20 on both sides gives, y ″ + 7 y ′ + 10 = 0 . The associated auxiliary equation is, r 2 + 7 r + 10 = 0 . Simplify r 2 + 7 r + 10 = 0 as, r 2 + 2 r + 5 r + 10 = 0 r ( r + 2 ) + 5 ( r + 2 ) = 0 ( r + 2 ) ( r + 5 ) = 0 r = − 2 , − 5 That is, the roots are r 1 = − 2 and r 2 = − 5 . Thus, the solution form y ( t ) = C 1 e r 1 t + C 2 e r 2 t becomes y ( t ) = C 1 e − 2 t + C 2 e − 5 t . Given that, the mass is displaced 25 cm to the right of the equilibrium point and a velocity of 1m/sec to the left. Convert 25cm to meters as 1 4 m . That is, y ( 0 ) = 1 4 and y ′ ( 0 ) = − 1 . Note that, the negative sign indicates to the left direction. Apply the initial conditions to obtain the value of the constants as follows: Substitute t = 0 and y ( 0 ) = 1 4 in y ( t ) = C 1 e − 2 t + C 2 e − 5 t , y ( 0 ) = C 1 e − 2 × 0 + C 2 e − 5 × 0 1 4 = C 1 ( 1 ) + C 2 ( 1 ) C 1 + C 2 = 1 4 C 1 = 1 4 − C 2 Differentiate y ( t ) = C 1 e − 2 t + C 2 e − 5 t with respect to t as, y ′ ( t ) = − 2 C 1 e − 2 t − 5 C 2 e − 5 t . Substitute t = 0 and y ′ ( 0 ) = − 1 in y ′ ( t ) = − 2 C 1 e − 2 t − 5 C 2 e − 5 t , y ′ ( 0 ) = − 2 C 1 e − 2 × 0 − 5 C 2 e − 5 × 0 − 1 = − 2 C 1 − 5 C 2 − 1 = − 2 ( 1 4 − C 2 ) − 5 C 2 ( ∵ C 1 = 1 4 − C 2 ) − 1 = − 1 2 + 2 C 2 − 5 C 2 Simplify further as, − 1 = − 1 2 − 3 C 2 3 C 2 = 1 2 C 2 = 1 6 Thus, C 1 = 1 4 − C 2 becomes, C 1 = 1 4 − 1 6 = 6 − 4 24 = 2 24 = 1 12 That is, C 1 = 1 12 and C 2 = 1 6 . Thus, the solution y ( t ) = C 1 e − 2 t + C 2 e − 5 t becomes, y ( t ) = 1 12 e − 2 t + 1 6 e − 5 t . The aim is to find the time at which the mass first return to its equilibrium position. Note that, the equilibrium position corresponds to y = 0 . That is, solve y ( t ) = 1 12 e − 2 t + 1 6 e − 5 t for y ( t ) = 0 . Note that, the value of the exponential is never zero. But, it tends to zero as t → ∞ . Thus, it is impossible for the object to reach the equilibrium position. Therefore, the mass never _ returns to its equilibrium position.

Solution Summary: The author explains how the governing equation of a mass-spring system is: cF_ext=left[inertiaright]y

Fundamentals of Differential Equations (9th Edition)

Fundamentals of Differential Equations (9th Edition)

9th Edition

ISBN: 9780321977069

Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider

Publisher: PEARSON

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Chapter 4.9, Problem 7E

Chapter 4.9, Problem 9E

Chapter 4 Solutions

Fundamentals of Differential Equations (9th Edition)

Ch. 4.1 - Verify that for b = 0 and Fext(t) = 0, equation...Ch. 4.1 - If Fext(t) = 0, equation (3) becomes my + by + ky...Ch. 4.1 - Show that if Fext(t) = 0, m = 1, k = 9, and b = 6,...Ch. 4.1 - Verify that y = sin 3t + 2 cos 3t is a solution to...Ch. 4.1 - Prob. 5E Ch. 4.1 - An external force F(t) = 2 cos 2t is applied to a...Ch. 4.1 - Prob. 7E Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - Prob. 10E

Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - Prob. 3E Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - Prob. 5E Ch. 4.2 - Prob. 6E Ch. 4.2 - Prob. 7E Ch. 4.2 - Prob. 8E Ch. 4.2 - Prob. 9E Ch. 4.2 - Prob. 10E Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - Prob. 15E Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - Prob. 17E Ch. 4.2 - Prob. 18E Ch. 4.2 - Prob. 19E Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - Prob. 21E Ch. 4.2 - Prob. 22E Ch. 4.2 - Prob. 23E Ch. 4.2 - Prob. 24E Ch. 4.2 - Prob. 25E Ch. 4.2 - Prob. 26E Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - Prob. 31E Ch. 4.2 - Prob. 32E Ch. 4.2 - Prob. 33E Ch. 4.2 - Prob. 34E Ch. 4.2 - Prob. 35E Ch. 4.2 - Using the definition in Problem 35, prove that if...Ch. 4.2 - Prob. 37E Ch. 4.2 - Prob. 38E Ch. 4.2 - Prob. 39E Ch. 4.2 - Prob. 40E Ch. 4.2 - In Problems 3741, find three linearly independent...Ch. 4.2 - Prob. 42E Ch. 4.2 - Prob. 43E Ch. 4.2 - Solve the initial value problem: y 2y y + 2y =...Ch. 4.2 - Prob. 45E Ch. 4.2 - Prob. 46E Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - Prob. 5E Ch. 4.3 - Prob. 6E Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 920, find a general solution. 9. y 8y...Ch. 4.3 - In Problems 920, find a general solution. 10. y +...Ch. 4.3 - Prob. 11E Ch. 4.3 - Prob. 12E Ch. 4.3 - In Problems 920, find a general solution. 13. y ...Ch. 4.3 - Prob. 14E Ch. 4.3 - Prob. 15E Ch. 4.3 - Prob. 16E Ch. 4.3 - In Problems 920, find a general solution. 17. y y...Ch. 4.3 - In Problems 920, find a general solution. 18. 2y +...Ch. 4.3 - Prob. 19E Ch. 4.3 - In Problems 920, find a general solution. 20. y y...Ch. 4.3 - In Problems 2127, solve the given initial value...Ch. 4.3 - Prob. 22E Ch. 4.3 - Prob. 23E Ch. 4.3 - In Problems 2127, solve the given initial value...Ch. 4.3 - Prob. 25E Ch. 4.3 - In Problems 2127, solve the given initial value...Ch. 4.3 - Prob. 27E Ch. 4.3 - Prob. 28E Ch. 4.3 - Prob. 29E Ch. 4.3 - Prob. 30E Ch. 4.3 - Prob. 31E Ch. 4.3 - Prob. 32E Ch. 4.3 - Prob. 33E Ch. 4.3 - Prob. 34E Ch. 4.3 - Swinging Door. The motion of a swinging door with...Ch. 4.3 - Prob. 36E Ch. 4.3 - Prob. 37E Ch. 4.3 - Prove the sum of angles formula for the sine...Ch. 4.4 - In Problems 18, decide whether or not the method...Ch. 4.4 - Prob. 2E Ch. 4.4 - Prob. 3E Ch. 4.4 - Prob. 4E Ch. 4.4 - Prob. 5E Ch. 4.4 - Prob. 6E Ch. 4.4 - Prob. 7E Ch. 4.4 - Prob. 8E Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - Prob. 16E Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - Prob. 18E Ch. 4.4 - Prob. 19E Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - Prob. 23E Ch. 4.4 - Prob. 24E Ch. 4.4 - Prob. 25E Ch. 4.4 - Prob. 26E Ch. 4.4 - Prob. 27E Ch. 4.4 - Prob. 28E Ch. 4.4 - Prob. 29E Ch. 4.4 - Prob. 30E Ch. 4.4 - Prob. 31E Ch. 4.4 - Prob. 32E Ch. 4.4 - Prob. 33E Ch. 4.4 - Prob. 34E Ch. 4.4 - Prob. 35E Ch. 4.4 - In Problems 3336, use the method of undetermined...Ch. 4.5 - Given that y1(t) = cos t is a solution to y y + y...Ch. 4.5 - Prob. 2E Ch. 4.5 - Prob. 3E Ch. 4.5 - Prob. 4E Ch. 4.5 - Prob. 5E Ch. 4.5 - Prob. 6E Ch. 4.5 - Prob. 7E Ch. 4.5 - In Problems 38, a nonhomogeneous equation and a...Ch. 4.5 - Prob. 9E Ch. 4.5 - Prob. 10E Ch. 4.5 - Prob. 11E Ch. 4.5 - Prob. 12E Ch. 4.5 - Prob. 13E Ch. 4.5 - In Problems 916 decide whether the method of...Ch. 4.5 - Prob. 15E Ch. 4.5 - Prob. 16E Ch. 4.5 - Prob. 17E Ch. 4.5 - Prob. 18E Ch. 4.5 - Prob. 19E Ch. 4.5 - In Problems 1722, find a general solution to the...Ch. 4.5 - Prob. 21E Ch. 4.5 - Prob. 22E Ch. 4.5 - Prob. 23E Ch. 4.5 - Prob. 24E Ch. 4.5 - Prob. 25E Ch. 4.5 - Prob. 26E Ch. 4.5 - Prob. 27E Ch. 4.5 - Prob. 28E Ch. 4.5 - Prob. 29E Ch. 4.5 - Prob. 30E Ch. 4.5 - In Problems 3136, determine the form of a...Ch. 4.5 - In Problems 3136, determine the form of a...Ch. 4.5 - Prob. 33E Ch. 4.5 - In Problems 3136, determine the form of a...Ch. 4.5 - Prob. 35E Ch. 4.5 - Prob. 36E Ch. 4.5 - Prob. 37E Ch. 4.5 - Prob. 38E Ch. 4.5 - In Problems 3740, find a particular solution to...Ch. 4.5 - Prob. 40E Ch. 4.5 - Prob. 41E Ch. 4.5 - Prob. 42E Ch. 4.5 - Prob. 43E Ch. 4.5 - Prob. 44E Ch. 4.5 - Prob. 46E Ch. 4.5 - Prob. 47E Ch. 4.5 - Prob. 48E Ch. 4.6 - Prob. 1E Ch. 4.6 - In Exercises 12, rectangles have been drawn to...Ch. 4.6 - Prob. 3E Ch. 4.6 - Prob. 4E Ch. 4.6 - Prob. 5E Ch. 4.6 - Prob. 6E Ch. 4.6 - Prob. 7E Ch. 4.6 - Prob. 8E Ch. 4.6 - Prob. 9E Ch. 4.6 - Prob. 10E Ch. 4.6 - Prob. 11E Ch. 4.6 - Prob. 12E Ch. 4.6 - Prob. 13E Ch. 4.6 - Prob. 14E Ch. 4.6 - Prob. 15E Ch. 4.6 - In Exercises 1318, use a calculator or a computer...Ch. 4.6 - Prob. 17E Ch. 4.6 - In Exercises 1318, use a calculator or a computer...Ch. 4.6 - For f(x) = 3x + 2, evaluate the Riemann sums: (a)...Ch. 4.6 - For g(x) = 4x 1, evaluate the Riemann sums: (a)...Ch. 4.6 - Prob. 21E Ch. 4.6 - Use Table 5.9 to evaluate the Riemann sums: Table...Ch. 4.6 - The graph of f(t) is in Figure 5.40. Which of the...Ch. 4.6 - Prob. 24E Ch. 4.6 - Prob. 25E Ch. 4.7 - In Problems 1 through 4, use Theorem 5 to discuss...Ch. 4.7 - In Problems 1 through 4, use Theorem 5 to discuss...Ch. 4.7 - Prob. 3E Ch. 4.7 - In Problems 1 through 4, use Theorem 5 to discuss...Ch. 4.7 - Prob. 5E Ch. 4.7 - In Problems 5 through 8, determine whether Theorem...Ch. 4.7 - Prob. 7E Ch. 4.7 - In Problems 5 through 8, determine whether Theorem...Ch. 4.7 - Prob. 9E Ch. 4.7 - In Problems 9 through 14, find a general solution...Ch. 4.7 - Prob. 11E Ch. 4.7 - In Problems 9 through 14, find a general solution...Ch. 4.7 - Prob. 13E Ch. 4.7 - In Problems 9 through 14, find a general solution...Ch. 4.7 - In Problems 15 through 18, find a general solution...Ch. 4.7 - In Problems 15 through 18, find a general solution...Ch. 4.7 - Prob. 17E Ch. 4.7 - Prob. 18E Ch. 4.7 - Prob. 19E Ch. 4.7 - Prob. 20E Ch. 4.7 - Prob. 21E Ch. 4.7 - Prob. 22E Ch. 4.7 - Prob. 23E Ch. 4.7 - Solve the following CauchyEuler equations by using...Ch. 4.7 - Prob. 25E Ch. 4.7 - Prob. 26E Ch. 4.7 - Prob. 27E Ch. 4.7 - Prob. 28E Ch. 4.7 - Prove that if y1 and y2 are linearly independent...Ch. 4.7 - Prob. 30E Ch. 4.7 - Prob. 31E Ch. 4.7 - Prob. 32E Ch. 4.7 - Prob. 33E Ch. 4.7 - Prob. 34E Ch. 4.7 - Prob. 35E Ch. 4.7 - Prob. 36E Ch. 4.7 - Prob. 37E Ch. 4.7 - Prob. 38E Ch. 4.7 - Prob. 39E Ch. 4.7 - Prob. 40E Ch. 4.7 - Prob. 41E Ch. 4.7 - In Problems 41 through 44, a differential equation...Ch. 4.7 - Prob. 43E Ch. 4.7 - Prob. 44E Ch. 4.7 - Prob. 45E Ch. 4.7 - Prob. 46E Ch. 4.7 - Prob. 47E Ch. 4.7 - Prob. 48E Ch. 4.7 - Prob. 49E Ch. 4.7 - Prob. 50E Ch. 4.7 - Prob. 51E Ch. 4.7 - Prob. 52E Ch. 4.8 - Prob. 1E Ch. 4.8 - Prob. 2E Ch. 4.8 - Try to predict the qualitative features of the...Ch. 4.8 - Prob. 4E Ch. 4.8 - Prob. 5E Ch. 4.8 - Prob. 6E Ch. 4.8 - Prob. 7E Ch. 4.8 - Prob. 8E Ch. 4.8 - Prob. 9E Ch. 4.8 - Prob. 10E Ch. 4.8 - Prob. 11E Ch. 4.8 - Use reduction of order to derive the solution...Ch. 4.8 - Prob. 13E Ch. 4.8 - Prob. 14E Ch. 4.8 - Prob. 15E Ch. 4.8 - Prob. 16E Ch. 4.8 - Prob. 17E Ch. 4.9 - Prob. 1E Ch. 4.9 - All problems refer to the massspring configuration...Ch. 4.9 - Prob. 3E Ch. 4.9 - Prob. 4E Ch. 4.9 - Prob. 5E Ch. 4.9 - Prob. 6E Ch. 4.9 - Prob. 7E Ch. 4.9 - Prob. 8E Ch. 4.9 - All problems refer to the massspring configuration...Ch. 4.9 - Prob. 10E Ch. 4.9 - Prob. 11E Ch. 4.9 - Prob. 12E Ch. 4.9 - Prob. 13E Ch. 4.9 - Prob. 14E Ch. 4.9 - Prob. 15E Ch. 4.9 - Prob. 16E Ch. 4.9 - Prob. 17E Ch. 4.9 - Prob. 18E Ch. 4.10 - In the following problems, take g = 32 ft/sec2 for...Ch. 4.10 - Prob. 2E Ch. 4.10 - Prob. 3E Ch. 4.10 - Prob. 4E Ch. 4.10 - Prob. 5E Ch. 4.10 - Prob. 6E Ch. 4.10 - Prob. 7E Ch. 4.10 - Prob. 8E Ch. 4.10 - In the following problems, take g = 32 ft/sec2 for...Ch. 4.10 - Prob. 10E Ch. 4.10 - Prob. 11E Ch. 4.10 - Prob. 12E Ch. 4.10 - In the following problems, take g = 32 ft/sec2 for...Ch. 4.10 - Prob. 14E Ch. 4.10 - Prob. 15E Ch. 4 - In Problems 128, find a general solution to the...Ch. 4 - Prob. 2RP Ch. 4 - Prob. 3RP Ch. 4 - Prob. 4RP Ch. 4 - Prob. 5RP Ch. 4 - Prob. 6RP Ch. 4 - Prob. 7RP Ch. 4 - Prob. 8RP Ch. 4 - Prob. 9RP Ch. 4 - Prob. 10RP Ch. 4 - Prob. 11RP Ch. 4 - Prob. 12RP Ch. 4 - Prob. 13RP Ch. 4 - Prob. 14RP Ch. 4 - Prob. 15RP Ch. 4 - Prob. 16RP Ch. 4 - Prob. 17RP Ch. 4 - In Problems 128, find a general solution to the...Ch. 4 - Prob. 19RP Ch. 4 - Prob. 20RP Ch. 4 - Prob. 21RP Ch. 4 - Prob. 22RP Ch. 4 - Prob. 23RP Ch. 4 - Prob. 24RP Ch. 4 - Prob. 25RP Ch. 4 - Prob. 26RP Ch. 4 - Prob. 27RP Ch. 4 - Prob. 28RP Ch. 4 - Prob. 29RP Ch. 4 - Prob. 30RP Ch. 4 - Prob. 31RP Ch. 4 - Prob. 32RP Ch. 4 - Prob. 33RP Ch. 4 - Prob. 34RP Ch. 4 - Prob. 35RP Ch. 4 - Prob. 36RP Ch. 4 - Prob. 37RP Ch. 4 - Prob. 38RP Ch. 4 - A 32-lb weight is attached to a vertical spring,...Ch. 4 - Prob. 1TWE Ch. 4 - Prob. 2TWE Ch. 4 - Prob. 3TWE Ch. 4 - For students with a background in linear algebra:...

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