Circular Motion using polar coordinates - Mechanics

In summary, a particle of mass m is constrained to slide on the inside of a vertical smooth semi-circular ring of radius r. The resultant force acting on the particle can be found using the expression F=ma(theta). Using conservation of energy, v2 can be expressed in terms of theta to solve for the resultant force.
  • #1
Keano16
23
0

Homework Statement



A particle of mass m is constrained to slide on the inside of a vertical smooth semi- circular ring of radius r. The position of the particle is described by a polar coordinate system whose origin is at the centre of the circle with axes along the orthogonal unit vectors r(hat) and θ(hat) where θ is the angle

Write down the resultant force acting on the particle as a function of θ.

Homework Equations



F=ma

The Attempt at a Solution



I have used the expression v = rθ(dot) to find that acceleration, a= rθ(double dot)θ(hat) −
v^2/r (rhat)
I'm having a problem getting the 'r' component of the accelration in terms of theta though, so that I can do F=ma(theta) to find resultant force.

Any help would be appreciated
 
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  • #2
The particle moves along a smooth vertical ring, so there is gravity, but no friction. You can use conservation of energy to get v2 in terms of theta.

ehild
 

FAQ: Circular Motion using polar coordinates - Mechanics

What is circular motion using polar coordinates?

Circular motion using polar coordinates is a way of describing the motion of an object moving in a circular path by using polar coordinates instead of Cartesian coordinates. This approach is useful for analyzing circular motion in physics and mechanics.

What is the difference between polar coordinates and Cartesian coordinates?

Polar coordinates use a distance from a central point and an angle from a reference direction to locate a point in space, while Cartesian coordinates use two perpendicular axes to locate a point in space. In circular motion, using polar coordinates can simplify calculations and better represent the circular path of an object.

How do you convert between polar and Cartesian coordinates?

To convert from polar to Cartesian coordinates, use the formulas x = r cos θ and y = r sin θ, where r is the distance from the origin and θ is the angle from the reference direction. To convert from Cartesian to polar coordinates, use the formulas r = √(x² + y²) and θ = tan⁻¹(y/x).

What is the equation for circular motion using polar coordinates?

The equation for circular motion using polar coordinates is r = r₀, where r is the distance from the origin, r₀ is the initial distance from the origin, and θ is the angular velocity. This equation represents a constant distance from the origin, but a changing angle over time, creating circular motion.

How is centripetal acceleration calculated using polar coordinates?

Centripetal acceleration in circular motion can be calculated in polar coordinates using the formula aᵣ = rω², where aᵣ is the radial acceleration, r is the distance from the origin, and ω is the angular velocity. This formula can be derived from the equations for velocity and acceleration in polar coordinates.

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