Physics Demonstrations

Double Spring

Mass 1 (kg)
Mass 2 (kg)
Outer Spring Constant (N/m)
Coupling Spring Constant (N/m)
Initial Displacement 1 (m)
Initial Displacement 2 (m)

In this demonstration, we observe a coupled spring-mass system: two masses ( $m_{1}$ , $m_{2}$ ) are connected to fixed walls by outer springs of constant $k$ , and to each other by a coupling spring of constant $k_{c}$ . The equations of motion are: $m_{1} {\ddot{x}}_{1} = - k x_{1} - k_{c} (x_{1} - x_{2})$ $m_{2} {\ddot{x}}_{2} = - k_{c} (x_{2} - x_{1}) - k x_{2}$ For equal masses $m_{1} = m_{2} = m$ , the system has two normal modes with angular frequencies: $ω_{1} = \sqrt{\frac{k}{m}} (in-phase mode)$ $ω_{2} = \sqrt{\frac{k + 2 k_{c}}{m}} (out-of-phase mode)$ In the in-phase mode both masses move together in the same direction; the coupling spring is neither stretched nor compressed. In the out-of-phase mode the masses move in opposite directions. For unequal masses or mixed initial conditions the motion appears as a superposition of both normal modes, producing energy transfer (beating) between the two masses. Some questions to consider while viewing the demonstration: Set both initial displacements equal and positive. Which normal mode does this excite? Set the initial displacements equal and opposite. Which normal mode does this excite? Set only one mass displaced. Describe how energy transfers between the masses over time. How does increasing the coupling spring constant $k_{c}$ affect the rate of energy transfer?