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A mass \(m_{1}\) rests on a frictionless horizontal table and is attached to a massless string. The string runs horizontally to the edge of the table, where it passes over a massless, frictionless pulley and then hangs vertically down. A second mass \(m_{2}\) is now attached to the bottom end of the string. Write down the Lagrangian for the system. Find the Lagrange equation of motion, and solve it for the acceleration of the blocks. For your generalized coordinate, use the distance \(x\) of the second mass below the tabletop.

Short Answer

Expert verified
The acceleration of the blocks is \(\ddot{x} = \frac{-m_2 g}{m_1 + m_2}\).

Step by step solution

01

Define the system and coordinates

We have two masses, \(m_1\) on a frictionless table and \(m_2\) hanging vertically. The string connects them over a pulley. We choose the distance \(x\) that mass \(m_2\) is below the tabletop as our generalized coordinate. As \(m_2\) moves vertically by \(x\), \(m_1\) moves horizontally by the same distance \(x\).
02

Write down the Kinetic Energy

The kinetic energy \(T\) of the system is the sum of the kinetic energies of both masses. Since \(m_1\) moves horizontally, its kinetic energy is \(\frac{1}{2} m_1 \dot{x}^2\), and since \(m_2\) moves vertically, its kinetic energy is \(\frac{1}{2} m_2 \dot{x}^2\). Therefore, the total kinetic energy is \(T = \frac{1}{2} m_1 \dot{x}^2 + \frac{1}{2} m_2 \dot{x}^2\).
03

Write down the Potential Energy

The potential energy \(U\) is due to the gravitational energy of \(m_2\). Since \(m_2\) is at a height \(x\), its potential energy is \(U = m_2 g x\). There is no potential energy for \(m_1\) as it moves horizontally on a frictionless table.
04

Formulate the Lagrangian

The Lagrangian \(L\) is defined as the difference between the kinetic and potential energies: \(L = T - U\). Substituting the expressions for \(T\) and \(U\), we get \(L = \frac{1}{2} m_1 \dot{x}^2 + \frac{1}{2} m_2 \dot{x}^2 - m_2 g x\).
05

Derive the Lagrange Equation of Motion

The Lagrange equation is given by \(\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) - \frac{\partial L}{\partial x} = 0\). First, compute \(\frac{\partial L}{\partial \dot{x}} = (m_1 + m_2) \dot{x}\). Then, compute \(\frac{\partial L}{\partial x} = -m_2 g\). Differentiating \(\frac{\partial L}{\partial \dot{x}}\) with respect to time gives \(\frac{d}{dt}\left((m_1 + m_2) \dot{x} \right) = (m_1 + m_2) \ddot{x}\). The Lagrange equation becomes \((m_1 + m_2) \ddot{x} + m_2 g = 0\).
06

Solve for the acceleration

Rearrange the equation \((m_1 + m_2) \ddot{x} = -m_2 g\) to solve for the acceleration \(\ddot{x}\). This gives \(\ddot{x} = \frac{-m_2 g}{m_1 + m_2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinetic Energy
Kinetic energy is the energy that an object possesses due to its motion. In our exercise with two blocks connected by a string over a pulley, both masses have kinetic energy as they move. For the block on the table, mass \(m_1\), which moves horizontally, its kinetic energy is given by the formula:\[T_{1} = \frac{1}{2} m_1 \dot{x}^2\]where \(\dot{x}\) represents the velocity of mass \(m_1\). For the second block, mass \(m_2\), which moves in the vertical direction, its kinetic energy is similarly given by:\[T_{2} = \frac{1}{2} m_2 \dot{x}^2\]Since both blocks move by the same amount due to the string connection, the total kinetic energy of the system is simply the sum of these individual kinetic energies:\[T = \frac{1}{2} m_1 \dot{x}^2 + \frac{1}{2} m_2 \dot{x}^2\]Understanding kinetic energy is important as it helps in formulating the total energy profile of a dynamic system like this one.
It is key to consider both contributions when analyzing the motion of the connected bodies.
Potential Energy
Potential energy relates to the position of an object in a gravitational field. In our problem, only mass \(m_2\) has potential energy because it is elevated above the ground. The formula for its potential energy is:\[U = m_2 g x\]Here, \(g\) is the acceleration due to gravity, which pulls the mass downward, and \(x\) is the height below the tabletop where mass \(m_2\) is located.
Mass \(m_1\), lying on a horizontal frictionless table, doesn't gain or lose height so doesn't possess gravitational potential energy in this setup.
The understanding of potential energy helps in determining how the stored energy can be converted into kinetic energy as the masses move, particularly when mass \(m_2\) descends and mass \(m_1\) is pulled across the tabletop. This concept is crucial for formulating the Lagrangian, where the difference between kinetic and potential energies helps us understand the dynamics of the system.
Equations of Motion
Equations of motion describe how the positions, velocities, or accelerations of a system change over time. Using Lagrangian mechanics, these equations result from finding a path that minimizes the action, a principle central to this formalism.1. **Formulating the Lagrangian:** The Lagrangian \(L\) is defined as the difference between the kinetic energy \(T\) and potential energy \(U\) of the system: \[ L = T - U = \frac{1}{2} m_1 \dot{x}^2 + \frac{1}{2} m_2 \dot{x}^2 - m_2 g x \]2. **Lagrange's Equation of Motion:** Derived using the Lagrangian, it is given by: \[ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) - \frac{\partial L}{\partial x} = 0 \] Applying this to our formulated Lagrangian gives us: \[ (m_1 + m_2) \ddot{x} + m_2 g = 0 \]3. **Solving for Acceleration:** Rearranging the equation leads to finding the acceleration \(\ddot{x}\): \[ \ddot{x} = \frac{-m_2 g}{m_1 + m_2} \]Understanding these equations allows us to predict how quickly the blocks will accelerate. Such equations are foundational to analyzing and projecting the motion of mechanical systems. They provide a deeper insight into how forces cause changes in motion and are crucial for designing systems in engineering and physics applications.

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Most popular questions from this chapter

A particle is confined to move on the surface of a circular cone with its axis on the \(z\) axis, vertex at the origin (pointing down), and half-angle \(\alpha\). The particle's position can be specified by two generalized coordinates, which you can choose to be the coordinates \((\rho, \phi)\) of cylindrical polar coordinates. Write down the equations that give the three Cartesian coordinates of the particle in terms of the generalized coordinates ( \(\rho, \phi\) ) and vice versa.

The "spherical pendulum" is just a simple pendulum that is free to move in any sideways direction. (By contrast a "simple pendulum"- unqualified - is confined to a single vertical plane.) The bob of a spherical pendulum moves on a sphere, centered on the point of support with radius \(r=R\) the length of the pendulum. A convenient choice of coordinates is spherical polars, \(r, \theta, \phi,\) with the origin at the point of support and the polar axis pointing straight down. The two variables \(\theta\) and \(\phi\) make a good choice of generalized coordinates. (a) Find the Lagrangian and the two Lagrange equations. (b) Explain what the \(\phi\) equation tells us about the \(z\) component of angular momentum \(\ell_{z^{*}}\) (c) For the special case that \(\phi=\) const, describe what the \(\theta\) equation tells us. (d) Use the \(\phi\) equation to replace \(\dot{\phi}\) by \(\ell_{z}\) in the \(\theta\) equation and discuss the existence of an angle \(\theta_{\mathrm{o}}\) at which \(\theta\) can remain constant. Why is this motion called a conical pendulum? (e) Show that if \(\theta=\theta_{0}+\epsilon,\) with \(\epsilon\) small, then \(\theta\) oscillates about \(\theta_{\mathrm{o}}\) in harmonic motion. Describe the motion of the pendulum's bob.

Two equal masses, \(m_{1}=m_{2}=m,\) are joined by a massless string of length \(L\) that passes through a hole in a frictionless horizontal table. The first mass slides on the table while the second hangs below the table and moves up and down in a vertical line. (a) Assuming the string remains taut, write down the Lagrangian for the system in terms of the polar coordinates \((r, \phi)\) of the mass on the table. (b) Find the two Lagrange equations and interpret the \(\phi\) equation in terms of the angular momentum \(\ell\) of the first mass. (c) Express \(\dot{\phi}\) in terms of \(\ell\) and eliminate \(\dot{\phi}\) from the \(r\) equation. Now use the \(r\) equation to find the value \(r=r_{0}\) at which the first mass can move in a circular path. Interpret your answer in Newtonian terms. (d) Suppose the first mass is moving in this circular path and is given a small radial nudge. Write \(r(t)=r_{0}+\epsilon(t)\) and rewrite the \(r\) equation in terms of \(\epsilon(t)\) dropping all powers of \(\epsilon(t)\) higher than linear. Show that the circular path is stable and that \(r(t)\) oscillates sinusoidally about \(r_{\mathrm{o}}\) What is the frequency of its oscillations?

Using the usual angle \(\phi\) as generalized coordinate, write down the Lagrangian for a simple pendulum of length \(l\) suspended from the ceiling of an elevator that is accelerating upward with constant acceleration \(a\). (Be careful when writing \(T\); it is probably safest to write the bob's velocity in component form.) Find the Lagrange equation of motion and show that it is the same as that for a normal, nonaccelerating pendulum, except that \(g\) has been replaced by \(g+a\). In particular, the angular frequency of small oscillations is \(\sqrt{(g+a) / l}\).

The center of a long frictionless rod is pivoted at the origin, and the rod is forced to rotate in a horizontal plane with constant angular velocity \(\omega\). Write down the Lagrangian for a bead threaded on the rod, using \(r\) as your generalized coordinate, where \(r, \phi\) are the polar coordinates of the bead. (Notice that \(\phi\) is not an independent variable since it is fixed by the rotation of the rod to be \(\phi=\omega t\).) Solve Lagrange's equation for \(r(t) .\) What happens if the bead is initially at rest at the origin? If it is released from any point \(r_{\mathrm{o}}>0,\) show that \(r(t)\) eventually grows exponentially. Explain your results in terms of the centrifugal force \(m \omega^{2} r\).

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