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A small cart (mass m) is mounted on rails inside a large
Chapter 7, Problem 7.23(choose chapter or problem)
A small cart (mass m) is mounted on rails inside a large cart. The two are attached by a spring (force constant k) in such a way that the small cart is in equilibrium at the midpoint of the large. The distance of the small cart from its equilibrium is denoted x and that of the large one from a fixed point on the ground is X, as shown in Figure 7.13. The large cart is now forced to oscillate such that \(X=A \cos \omega t\), with both A and \(\omega\) fixed. Set up the Lagrangian for the motion of the small cart and show that the Lagrange equation has the form
\(\ddot{x}+\omega_{0}^{2} x=B \cos \omega t\)
where \(\omega_{0}\) is the natural frequency \(\omega_{\mathrm{o}}=\sqrt{k / m}\) and B is a constant. This is the form assumed in Section 5.5, Equation (5.57), for driven oscillations (except that we are here ignoring damping). Thus the system
described here would be one way to realize the motion discussed there. (We could fill the large cart with molasses to provide some damping.)
Questions & Answers
QUESTION:
A small cart (mass m) is mounted on rails inside a large cart. The two are attached by a spring (force constant k) in such a way that the small cart is in equilibrium at the midpoint of the large. The distance of the small cart from its equilibrium is denoted x and that of the large one from a fixed point on the ground is X, as shown in Figure 7.13. The large cart is now forced to oscillate such that \(X=A \cos \omega t\), with both A and \(\omega\) fixed. Set up the Lagrangian for the motion of the small cart and show that the Lagrange equation has the form
\(\ddot{x}+\omega_{0}^{2} x=B \cos \omega t\)
where \(\omega_{0}\) is the natural frequency \(\omega_{\mathrm{o}}=\sqrt{k / m}\) and B is a constant. This is the form assumed in Section 5.5, Equation (5.57), for driven oscillations (except that we are here ignoring damping). Thus the system
described here would be one way to realize the motion discussed there. (We could fill the large cart with molasses to provide some damping.)
ANSWER:Step 1 of 5
The following are given by the question:
The small cart is connected with spring that has spring constant k. The small cart has an equilibrium position at the midpoint of the large cart.
X is the distance between the the fixed point and midpoint of large cart and x is the distance of the small cart from the equilibrium point.
The kinetic energy of the small cart
\(T=\frac{1}{2} m v^{2}\) (1)
The potential energy of the system
\(U=\frac{1}{2} k x^{2}\)