Polar Coordinates and Motion

Questions and Answers

What does radial acceleration represent in plane polar coordinates?

Acceleration due to a constant angular velocity

Acceleration due to a change in tangential speed

The acceleration directed outward from the center of motion

Acceleration acting towards the center of the circular path (correct)

Which component contributes to the Coriolis acceleration when both r and θ change over time?

Constant angular velocity

Tangential speed change (correct)

Radial velocity change (correct)

Centripetal acceleration

In polar coordinates, what does Newton’s law imply about the forces acting in different directions?

Tangential acceleration is always zero in polar coordinates

Forces in polar coordinates are equivalent to those in Cartesian coordinates

Acceleration must always be directed outward from the center

The form of Newton's law varies across different coordinate systems (correct)

What condition must be met for the Coriolis acceleration to be present?

Both r and θ must be changing over time Signup and view all the answers

What type of motion does a fixed radial distance with a varying angular position imply?

Motion along the arc of a circle Signup and view all the answers

What results when the tangential speed vθ changes due to a change in radial distance Δr?

A contribution to the tangential acceleration occurs Signup and view all the answers

What does the centripetal acceleration act upon during motion in a circular path?

Inward towards the center of the circular path Signup and view all the answers

When analyzing the motion in plane polar coordinates, which aspect is NOT covered by the acceleration formula?

Constant linear acceleration Signup and view all the answers

How is the x-coordinate of the center of mass calculated for a system of particles?

$x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$ Signup and view all the answers

In a 3D particle system, what does the position vector of the center of mass represent?

The average position weighted by mass Signup and view all the answers

For a two-particle system positioned at $(x_1, y_1)$ and $(x_2, y_2)$, how would you express the y-coordinate of the center of mass?

$y_{cm} = \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2}$ Signup and view all the answers

When considering a system of n particles, which expression is correct for the z-coordinate of the center of mass?

$z_{cm} = \frac{\sum m_i z_i}{\sum m_i}$ Signup and view all the answers

If you have a 3-particle system, which formula correctly represents the x-coordinate of the center of mass?

$x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$ Signup and view all the answers

What does the position vector in polar coordinates represent?

The position in terms of radial distance and angle Signup and view all the answers

What is the primary characteristic of velocity in plane polar coordinates?

It consists of both radial and tangential components Signup and view all the answers

In the context of polar coordinates, which variable indicates the radial distance?

r Signup and view all the answers

What is the expression for the velocity vector in polar coordinates?

$rac{d r}{dt} r̂ + r rac{d heta}{dt} hetâ$ Signup and view all the answers

When is the radial velocity component constant in polar coordinates?

When r varies and θ is constant Signup and view all the answers

In the given equations, what does the notation $ṙ$ signify?

Rate of change of radial distance Signup and view all the answers

Which of the following correctly describes tangential velocity in polar coordinates?

Related to the rate of change of the angle Signup and view all the answers

Which scenario qualifies as a combination of radial and tangential motion?

An object accelerating away from a fixed point while rotating Signup and view all the answers

What are the coordinates used in the Cartesian coordinate system?

(x, y) Signup and view all the answers

Which coordinate system is described as suitable for circular motion?

Plane polar coordinate system Signup and view all the answers

In the plane polar coordinate system, what does the variable 'r' represent?

Radial distance from the origin Signup and view all the answers

How is the position of a point represented in the plane polar coordinate system?

P(r, θ) Signup and view all the answers

Which coordinate system has three dimensions?

Cylindrical coordinate system Signup and view all the answers

In Cartesian coordinates, how are lines of constant x and y represented?

Perpendicular lines Signup and view all the answers

What does the angle θ represent in the plane polar coordinate system?

The angle from a fixed direction (X-axis) Signup and view all the answers

Which of the following is NOT a coordinate system mentioned?

Square coordinate system Signup and view all the answers

What is the significance of the center of mass being independent of the origin in a system of particles?

It indicates that the motion of the system does not depend on the position of the observer. Signup and view all the answers

In terms of mass elements, how is the position of the center of mass calculated?

By taking the average of the positions weighted by the mass of each element. Signup and view all the answers

What happens to the moment of masses about the origin if the center of mass is chosen as the origin?

The moment of masses becomes zero. Signup and view all the answers

As N approaches infinity in the continuum limit, how is the expression for the center of mass derived?

By treating the mass elements as continuous variables and integrating. Signup and view all the answers

Which of the following equations represents the moment of masses about the origin?

$\Sigma m_i r_i = 0$ for a balanced system. Signup and view all the answers

Study Notes

Motion in Polar Coordinates

Position vector in polar coordinates is represented as ( \vec{r} = r \hat{e_r} ).
Two cases of motion:
- Radial motion: constant angle ( \theta ), ( \vec{v} = \dot{r} \hat{e_r} ).
- Circular motion: fixed radius ( r ), varying angle ( \theta ), ( \vec{v} = r \dot{\theta} \hat{e_\theta} ).

Acceleration in Polar Coordinates

Acceleration in polar coordinates is the sum of radial and tangential acceleration.
The full expression for acceleration: [ \vec{a} = \ddot{r} \hat{e_r} + r \dot{\theta}^2 \hat{e_r} + (r \ddot{\theta} + 2 \dot{r} \dot{\theta}) \hat{e_\theta} ]
Coriolis acceleration occurs when both ( r ) and ( \theta ) vary with time.

Coriolis Acceleration

Split into components based on changes in radial velocity and tangential speed.
Half is due to radial velocity change, and half is related to tangential speed variations.

Newton’s Law in Polar Coordinates

Newton's laws are not in Cartesian form in polar coordinates.
For radial direction: [ F_r = m(\ddot{r} - r \dot{\theta}^2) ]
For tangential direction: [ F_\theta = m(r \ddot{\theta} + 2 \dot{r} \dot{\theta}) ]

Center of Mass in 2D

For two particles not along the x-axis: [ x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}, \quad y_{cm} = \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2} ]
Important for understanding motion and distribution of mass.

Multi-Particle System Center of Mass

Center of mass for a system of ( n ) particles in 3D: [ x_{cm} = \frac{\sum m_i x_i}{\sum m_i}, \quad y_{cm} = \frac{\sum m_i y_i}{\sum m_i}, \quad z_{cm} = \frac{\sum m_i z_i}{\sum m_i} ]

Various Coordinate Systems

Common coordinate systems include:
- Cartesian: defined by ( (x, y) ).
- Plane Polar: defined by ( (r, \theta) ).
- Cylindrical: ( (\rho, \phi, z) ).
- Spherical: ( (r, \theta, \phi) ).
Cartesian coordinates are frequently used for linear motion.

Relation Between Cartesian and Polar Coordinates

Position in polar coordinates relates to Cartesian as follows: [ x = r \cos(\theta), \quad y = r \sin(\theta) ]

Motion Analysis in Plane Polar Coordinates

Velocity has both radial and tangential components: [ \vec{v} = \dot{r} \hat{e_r} + r \dot{\theta} \hat{e_\theta} ]
Significant when analyzing systems moving in circular paths.

Pseudo Forces in Polar Motion

In non-inertial frames, a pseudo force is added to modify Newton's laws for motion in polar dynamics.

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Description

This quiz covers the concepts of motion in plane polar coordinates, focusing on both radial and tangential velocities. It delves into the implications of constant radius and varying angles, emphasizing the acceleration associated with these movements. Test your understanding of these fundamental principles in physics.