Circular accelaration with increasing velocity?

[Denken]

circular accelaration with increasing velocity?

Homework Statement

Object starts at rest at an angle of 0 degree's. the object is accelerated along a 45 degree arc with a radius of 2 meters to a speed of 7m/s at which point it is released. the object has a mass of .3 Kg. What is the equation for the force applied along the arc?

Homework Equations

circular accel. = V[2][/r] and other kinematics and derivatives and anti-derivatives

The Attempt at a Solution

way more work than i am going to put on here, but in the end i got a jerk function of 960.4/pi
... not fun

 

[americanforest]

To keep it in a circle you need a certain force. To accelerate it tangentially you need another. Superpose them to get the total force.

$$\vec{F}=-\frac{mv^{2}}{R}\hat{\rho}+F_{tan}\hat{\theta}$$

Then use basic kinematic equations with the information given to find the tangential force. Then it's just a matter of getting the magnitude, which we expect to be time dependent due to the centripetal contribution.

 

[Denken]

just a couple questions on that ... what does the {Rho} represent and why is it Ftan?

 

[americanforest]

The rho is the radial unit vector. Centrifugal forces push the mass out; in order for it to stay in a circle there needs to be an equal and opposite force pulling it in. This is what the force in the negative rho direction represents.

Mathematically,

$$\hat{\rho}=cos(\theta)\hat{x}+sin(\theta)\hat{y}$$

As for the other component of the force it is directed tangentially to the circle because it increases the tangential speed of the mass.