Question

In designing circular rides for amusement parks, mechanical engineers must consider how small variations in certain parameters can alter the net force on a passenger. Consider a passenger of mass m riding around a horizontal circle of radius rat speed v. What is the variation dF in the net force magnitude for(a) a variation dr in the radius with ν held constant,(b) a variation dv in the speed with r held constant, and(c) a variation dT in the period with r held constant?

Answer 1

Part a)

`dF = -(mv^2)/(r^2) dr`

Part b)

`dF = (2mvdv)/(r)`

Part c)

`dT = - (2\pi r)/(v^2) dv`

Step-by-step explanation:

Part a)

As we know that force on the passenger while moving in circle is given as

`F = (mv^2)/(r)`

now variation in force is given as

`dF = -(mv^2)/(r^2) dr`

here speed is constant

Part b)

Now if the variation in force is required such that r is constant then we will have

`F = (mv^2)/(r)`

so we have

`dF = (2mvdv)/(r)`

Part c)

As we know that time period of the circular motion is given as

`T = (2\pi r)/(v)`

so here if radius is constant then variation in time period is given as

`dT = - (2\pi r)/(v^2) dv`