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The result you got above should be x(t) = e(C + Dt), eq 77. This introduces a new solution to the differential equation of motion, namely x(t) = Dte. Show directly that this is a solution of eq. 47 for y/2 = Mo-

The result you got above should be x(t) = e(C + Dt), eq 77. This introduces a new solution to the differential equation of motion, namely x(t) = Dte. Show directly that this is a solution of eq. 47 for y/2 = Mo-

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The result you got above should be \( x(t) = e^{-\frac{\gamma t}{2}} (C + Dt) \), eq 77. This introduces a new solution to the differential equation of motion, namely \( x(t) = Dte^{-\frac{\gamma t}{2}} \). Show directly that this is a solution of eq. 47 for \(\gamma/2 = \omega_0\).
Transcribed Image Text:The result you got above should be \( x(t) = e^{-\frac{\gamma t}{2}} (C + Dt) \), eq 77. This introduces a new solution to the differential equation of motion, namely \( x(t) = Dte^{-\frac{\gamma t}{2}} \). Show directly that this is a solution of eq. 47 for \(\gamma/2 = \omega_0\).
The image contains a set of equations and explanatory text related to damping forces in physics. Here's the transcription: --- **\( F_{\text{damping}} = -b \dot{x}. \) \hfill (46)** Note that this is *not* the force from sliding friction on a table. That would be a force with constant magnitude \(\mu_k N\). The \(-b \dot{x}\) force here pertains to a body moving through a fluid, provided that the velocity isn’t too large. So it is in fact a realistic force. The \( F = ma \) equation for the mass is \[ F_{\text{spring}} + F_{\text{damping}} = m \ddot{x} \] \[ \Longrightarrow -kx - b \dot{x} = m \ddot{x} \] \[ \Longrightarrow \ddot{x} + \gamma \dot{x} + \omega_0^2 x = 0, \quad \text{where} \quad \omega_0 \equiv \sqrt{\frac{k}{m}}, \quad \gamma \equiv \frac{b}{m}. \hfill (47) \] --- This text explains that the damping force \(-b \dot{x}\) is different from sliding friction, as it's relevant for motion through a fluid. The equation of motion for the mass includes both spring and damping forces. Parameters \(\omega_0\) and \(\gamma\) are defined in terms of \(k\), \(m\), and \(b\).
Transcribed Image Text:The image contains a set of equations and explanatory text related to damping forces in physics. Here's the transcription: --- **\( F_{\text{damping}} = -b \dot{x}. \) \hfill (46)** Note that this is *not* the force from sliding friction on a table. That would be a force with constant magnitude \(\mu_k N\). The \(-b \dot{x}\) force here pertains to a body moving through a fluid, provided that the velocity isn’t too large. So it is in fact a realistic force. The \( F = ma \) equation for the mass is \[ F_{\text{spring}} + F_{\text{damping}} = m \ddot{x} \] \[ \Longrightarrow -kx - b \dot{x} = m \ddot{x} \] \[ \Longrightarrow \ddot{x} + \gamma \dot{x} + \omega_0^2 x = 0, \quad \text{where} \quad \omega_0 \equiv \sqrt{\frac{k}{m}}, \quad \gamma \equiv \frac{b}{m}. \hfill (47) \] --- This text explains that the damping force \(-b \dot{x}\) is different from sliding friction, as it's relevant for motion through a fluid. The equation of motion for the mass includes both spring and damping forces. Parameters \(\omega_0\) and \(\gamma\) are defined in terms of \(k\), \(m\), and \(b\).
Expert Solution
Step 1

We have a second-order differential equation as:

 x..+γx.+ωo2x=0

We have to prove that xt=Dte-γ2t is a solution of the above differential equation given that ωo=γ2

 

 

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