A mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c). The mass is set in motion with initial position x and initial velocity vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the form x(t) = C₁ e Pt cos (w₁t-α₁). Also, find the undamped position function u(t) = Cocos (-o) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs of x(t) and u(t). m=- ,C= 15 2 k=18, x = 6, Vo = 0 x(t)=, which means the system is ▼ (Use integers rdecimals for any numbers in the expression. Round to four decimal places as needed. Type any angle measures in radians. Use angle measures greater than or equal to 0 and less than or equal to 2x.) The undamped position is u(t)= (Use integers or decimals for any numbers in the expression. Round to four decimal places as needed. Type any angle measures in radians. Use angle measures greater than or equal to 0 and less than or equal to 2x.) Choose the correct graph that compares x(t) and u(t). OA. О в. О с. O D. Q M Wi Q JAA M
A mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c). The mass is set in motion with initial position x and initial velocity vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the form x(t) = C₁ e Pt cos (w₁t-α₁). Also, find the undamped position function u(t) = Cocos (-o) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs of x(t) and u(t). m=- ,C= 15 2 k=18, x = 6, Vo = 0 x(t)=, which means the system is ▼ (Use integers rdecimals for any numbers in the expression. Round to four decimal places as needed. Type any angle measures in radians. Use angle measures greater than or equal to 0 and less than or equal to 2x.) The undamped position is u(t)= (Use integers or decimals for any numbers in the expression. Round to four decimal places as needed. Type any angle measures in radians. Use angle measures greater than or equal to 0 and less than or equal to 2x.) Choose the correct graph that compares x(t) and u(t). OA. О в. О с. O D. Q M Wi Q JAA M
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:**Oscillatory Motion with Damping**
A mass is attached to both a spring (with given spring constant \( k \)) and a dashpot (with given damping constant \( c \)). The mass is set in motion with initial position \( x_0 \) and initial velocity \( v_0 \). Our objective is to find the position function \( x(t) \) and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, we will write the position function in the form \( x(t) = C_1 e^{-pt} \cos (\omega_1 t - \alpha_1) \). Additionally, we will find the undamped position function \( u(t) = C_0 \cos (\omega_0 t - \alpha_0) \) that would result if the mass on the spring were set in motion with the same initial position and velocity but with the dashpot disconnected (so \( c = 0 \)). Finally, we will visually illustrate the effect of damping by comparing the graphs of \( x(t) \) and \( u(t) \).
Given parameters:
- \( m = \frac{3}{4} \)
- \( c = \frac{15}{2} \)
- \( k = 18 \)
- \( x_0 = 6 \)
- \( v_0 = 0 \)
**Calculations and Graph Selection**
1. **Position Function \( x(t) \):**
- Assess the damping condition and compute the corresponding position function \( x(t) \).
- (Use integers or decimals for any numbers in the expression. Round to four decimal places as needed. Type any angle measures in radians. Use angle measures greater than or equal to 0 and less than or equal to \( 2\pi \).)
2. **Undamped Position Function \( u(t) \):**
- Calculate the undamped position function \( u(t) \).
- (Use integers or decimals for any numbers in the expression. Round to four decimal places as needed. Type any angle measures in radians. Use angle measures greater than or equal to 0 and less than or equal to \( 2\pi \).)
3. **Graph Comparison:**
- Choose the correct graph that compares \( x(t) \) and \( u(t) \) from the given graphs (Labelled A
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