Explanation Given: The mass of the bullet is m . The velocity of the bullet is v 0 . The mass of the target is M . The number of springs, n = 2 . Conclusion: Express the equivalent spring stiffness. k e q = n k (I) Substitute 2 for n in Equation (I). k e q = 2 k Express the total mass when bullet got embedded in the target. m T = m + M Express the natural frequency. ω n = k e q m T (II) Substitute 2 k for k e q and m + M for m T in Equation (II). ω n = 2 k m + M Express the equation describing the critically dampened system. x = C e − ( c / 2 m T ) t sin ( ω d t + ϕ ) (III) Here, damping natural frequency is ω d , time is t and phase angle is ϕ . Apply the initial condition: At, t = 0 , x = 0 Substitute 0 for x and 0 for t in Equation (III). 0 = C e − ( c / 2 m T ) ( 0 ) sin ( ω d × 0 + ϕ ) 0 = C e − 0 sin ϕ C sin ϕ = 0 As the C = 0 and sin ϕ = 0 take ϕ = 0 Substitute 0 for ϕ in Equation (III). x = C e − ( c / 2 m T ) t sin ( ω d t + 0 ) = C e − ( c / 2 m T ) t sin ( ω d t ) (IV) Take time derivative of Equation (IV) to get, x ˙ = v = C [ ω d e − ( c / 2 m T t ) − c 2 m T e − ( c / 2 m T t ) sin ω d t ] = C e − ( c / 2 m T t ) [ ω d cos ω d t − c 2 m T sin ω d t ] (V) As the linear momentum is conserved along the horizontal direction during the impact, thus write the expression as, + ← m v 0 = m T v (VI) Here, initial velocity is v 0 . Substitute m + M for m T in Equation (VI). + ← m v 0 = ( m + M ) v v = [ m m + M ] v 0 (VII) Substitute 0 for t and [ m m + M ] v 0 for v in Equation (V)

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Free Damped Vibrations

Free vibration is a type of vibration in which the external force is removed after giving the initial displacement to a body/system. In other words, in this type of vibration, force is applied at the initial displacement of the system, and after the init…

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Chapter 22.6, Problem 75P

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The spring’s maximum compression.

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Chapter 22.6, Problem 74P

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Chapter 22 Solutions

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Ch. 22.1 - A spring is stretched 175 mm by an 8-kg block. If...Ch. 22.1 - Prob. 2P Ch. 22.1 - A spring is stretched 200 mm by a 15-kg block. If...Ch. 22.1 - When a 20-lb weight is suspended from a spring,...Ch. 22.1 - Prob. 5P Ch. 22.1 - Prob. 6P Ch. 22.1 - Prob. 7P Ch. 22.1 - Prob. 8P Ch. 22.1 - A 3-kg block is suspended from a spring having a...Ch. 22.1 - Prob. 10P

Ch. 22.1 - Prob. 11P Ch. 22.1 - 22-12. Determine the natural period of vibration...Ch. 22.1 - The body of arbitrary shape has a mass m, mass...Ch. 22.1 - Determine the torsional stiffness k, measured in...Ch. 22.1 - Prob. 15P Ch. 22.1 - Prob. 16P Ch. 22.1 - If the natural periods of oscillation of the...Ch. 22.1 - Prob. 18P Ch. 22.1 - Prob. 19P Ch. 22.1 - A uniform board is supported on two wheels which...Ch. 22.1 - If the wire AB is subjected to a tension of 20 lb,...Ch. 22.1 - The bar has a length l and mass m. It is supported...Ch. 22.1 - The 20-kg disk, is pinned at its mass center O and...Ch. 22.1 - Prob. 24P Ch. 22.1 - If the disk in Prob. 22-24 has a mass of 10 kg,...Ch. 22.1 - Prob. 26P Ch. 22.1 - Prob. 27P Ch. 22.1 - Prob. 28P Ch. 22.1 - Prob. 29P Ch. 22.2 - Determine the differential equation of motion of...Ch. 22.2 - Determine the natural period of vibration of the...Ch. 22.2 - Determine the natural period of vibration of the...Ch. 22.2 - Prob. 33P Ch. 22.2 - Determine the differential equation of motion of...Ch. 22.2 - Prob. 35P Ch. 22.2 - Prob. 36P Ch. 22.2 - Prob. 37P Ch. 22.2 - Prob. 38P Ch. 22.2 - Prob. 39P Ch. 22.2 - If the slender rod has a weight of 5 lb, determine...Ch. 22.6 - If the block-and-spring model is subjected to the...Ch. 22.6 - Prob. 42P Ch. 22.6 - A 4-lb weight is attached to a spring having a...Ch. 22.6 - Prob. 44P Ch. 22.6 - Prob. 45P Ch. 22.6 - Prob. 46P Ch. 22.6 - Prob. 47P Ch. 22.6 - Prob. 48P Ch. 22.6 - Prob. 49P Ch. 22.6 - Prob. 50P Ch. 22.6 - The 40-kg block is attached to a spring having a...Ch. 22.6 - The 5kg circular disk is mounted off center on a...Ch. 22.6 - Prob. 53P Ch. 22.6 - Prob. 54P Ch. 22.6 - Prob. 55P Ch. 22.6 - Prob. 56P Ch. 22.6 - Prob. 57P Ch. 22.6 - Prob. 58P Ch. 22.6 - Prob. 59P Ch. 22.6 - The 450-kg trailer is pulled with a constant speed...Ch. 22.6 - Prob. 61P Ch. 22.6 - Prob. 62P Ch. 22.6 - Prob. 63P Ch. 22.6 - The spring system is connected to a crosshead that...Ch. 22.6 - Prob. 65P Ch. 22.6 - Prob. 66P Ch. 22.6 - Prob. 67P Ch. 22.6 - The 200-lb electric motor is fastened to the...Ch. 22.6 - Prob. 69P Ch. 22.6 - If two of these maximum displacements can be...Ch. 22.6 - Prob. 71P Ch. 22.6 - Prob. 72P Ch. 22.6 - Prob. 73P Ch. 22.6 - Prob. 74P Ch. 22.6 - Prob. 75P Ch. 22.6 - Prob. 76P Ch. 22.6 - Prob. 77P Ch. 22.6 - Prob. 78P Ch. 22.6 - Prob. 79P

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