Physics Chapter: Angular Momentum & Atwood's Machine

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Questions and Answers

What is the condition for the conservation of linear momentum in a system?

The total mass of the system is constant.

The velocity of all particles is equal.

The total force acting on the system is zero. (correct)

The system is in a state of rest.

How is the angular momentum of a particle defined?

It is the cross product of the radius vector and linear momentum. (correct)

It is the moment of inertia times angular velocity.

It is the product of mass and velocity.

It is the sum of all forces acting on the particle.

What happens to the angular momentum of a particle if the total torque is zero?

It changes unpredictably.

It increases proportionally to time.

It decreases over time.

It remains constant. (correct)

What is the relationship between work done by a force and the change in kinetic energy?

Work done is equal to the change in kinetic energy. Signup and view all the answers

Which of the following equations represents the definition of torque?

Torque = Radius x Force Signup and view all the answers

When is the conservation theorem for momentum applicable?

Only in inertial frames. Signup and view all the answers

Which of the following correctly states the relationship between linear momentum and force?

Force is the rate of change of momentum. Signup and view all the answers

Under what condition does the work done by a force equal the change in mechanical energy of a system?

When no non-conservative forces are present. Signup and view all the answers

What does the total angular momentum of a system around a reference point depend on?

The radius vector and the particles' velocities Signup and view all the answers

In the expression for total angular momentum 𝐿, what does the term $\sum r_i' \times p_i'$ represent?

The individual angular momentum of particles relative to the center of mass Signup and view all the answers

What happens to the total angular momentum equation when $\sum m_i r_i' = 0$?

The total angular momentum simplifies significantly Signup and view all the answers

What is the significance of the term $R \times Mv$ in the total angular momentum?

It accounts for the motion of the center of mass Signup and view all the answers

Which of the following is true concerning the angular momentum of a system of particles?

Angular momentum is conserved in an isolated system regardless of external forces Signup and view all the answers

In terms of particle velocities, which relation is highlighted for angular momentum calculation?

$v_i = v_{cm} + v_i'$ Signup and view all the answers

What role does the center of mass play in the determination of angular momentum?

It acts as a point through which all masses can be assumed to act Signup and view all the answers

Which of the following statements about total angular momentum L is accurate?

L includes contributions from both the motion of the center of mass and the individual particles Signup and view all the answers

What does the left side of the described equation represent?

The time derivative of angular momentum Signup and view all the answers

In the context of Atwood’s machine, what is the significance of the variable 𝑙?

It is the length of the rope between the weights Signup and view all the answers

What does the equation $𝑥̇ = \frac{𝑔 (𝑀_1 - 𝑀_2)}{𝑀_1 + 𝑀_2}$ represent in the context of the system?

The velocity of one mass in the Atwood's machine Signup and view all the answers

Which expression correctly represents the kinetic energy $T$ in Atwood's machine configuration?

T = $\frac{1}{2} (M_1 + M_2) ẋ^2$ Signup and view all the answers

In the context of Lagrangian mechanics, what does the term $L = T - V$ signify?

It denotes the Lagrangian of the system Signup and view all the answers

What condition must be met for a system like Atwood's machine to be considered conservative?

The system must have holonomic constraints Signup and view all the answers

What is the significance of the derivatives $\frac{\partial L}{\partial x}$ and $\frac{\partial L}{\partial \dot{x}}$ in the context of Lagrangian mechanics?

They are used to derive the equations of motion Signup and view all the answers

Which equation represents the conservation of angular momentum?

$L = I \omega$ Signup and view all the answers

What is the nature of the system when the pulley in Atwood's machine is assumed to be frictionless and massless?

Holonomic and conservative Signup and view all the answers

How does one define the position vector for the center of mass in a system with mass $M$?

$R = \frac{\sum m_i r_i}{M}$ Signup and view all the answers

Study Notes

The θ equation

The time derivative of angular momentum is equal to the applied torque.
The equation represents the conservation of angular momentum: the change in angular momentum over time equals the applied torque.
The right side of the equation is the torque, the left side is the rate of change of angular momentum.

Atwood's Machine

This example describes a conservative system with holonomic, scleronomous constraints.
The pulley is assumed frictionless and massless.
There is one independent coordinate, 'x', which represents the position of one of the weights.
The position of the other weight is determined by the constraint that the length of the rope between them is 'l'.
The potential energy 'V' is determined by the gravitational potential energy of the two masses.
The kinetic energy 'T' is determined by the velocity of the masses.
The Lagrangian is the difference between the kinetic and potential energy.
The equation of motion is derived by applying the Euler-Lagrange equation to the Lagrangian, resulting in an expression for the acceleration of the masses.

Problem 1

The formula for the magnitude of the position vector of the center of mass from an arbitrary origin is provided.
This formula involves a weighted sum of the positions of individual particles and the squared distance between them.

Problem 2

The Lagrange's equations can be expressed in the Nelson form, which is an alternative representation.

Angular Momentum

The total angular momentum of a system is the sum of the angular momenta of individual particles.
The angular momentum can be expressed in terms of the radius vectors from the origin to the center of mass and to individual particles, as well as their velocities.
The total angular momentum can be separated into two components: the angular momentum of the motion concentrated at the center of mass and the angular momentum due to motion about the center of mass.

Conservation of Linear Momentum

If the total force acting on a particle is zero, its linear momentum is conserved.
This means the linear momentum remains constant in time.

Angular Momentum of a Particle

The angular momentum of a particle is the cross product of its radius vector and its linear momentum.
The torque is defined as the cross product of the radius vector and the force acting on the particle.
The rate of change of angular momentum is equal to the torque applied to the particle.
If the total torque acting on a particle is zero, its angular momentum is conserved.
This means the angular momentum remains constant in time.

Work Done by a Force

Work done by a force on a particle is the integral of the dot product of the force and the displacement.
The work done by a constant force is calculated by multiplying the magnitude of the force with the displacement.

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Description

Explore the fundamental concepts of angular momentum and the mechanics of Atwood's Machine in this quiz. Understand the relationship between torque and angular momentum as well as the energy dynamics within a conservative system. Test your knowledge of these topics through various questions on classical mechanics.