Vector Problem: Acceleration, Velocity & Arc Length

[nns91]

Homework Statement

A slender rod through the origin of the polar plane rotates about the origin at the rate of 3 rad/min. A beetle starting from the point (2,) crawls along the rod toward the origin at the rate of 1 in/min. The axes are marked in inches

a. Find the beetle's acceleration and velocity in polar form when it is halfway to (1 inch from) the origin.
b.To the nearest thousandth of an inch, what will be the length of the path the beetle has traveled by the time it reaches the origin.

Homework Equations

acceleration, velocity, arc length

The Attempt at a Solution

Can anyone hint me how to do this problem ?

For part b, I guess I just take the integral of speed from 0 to t in which t is the time the beetle needs to reach the origin ??

 

[Dr.D]

To begin the problem,
r = mag(r) er
where r is the radius vector to the bug, mag(r) is the length of r, and er is a unit vector in the direction of r. Then differentiate this vector expression, remembering that the derivative of er is theta-dot etheta, and the derivative of etheta is -thetadot er.

That should get you started.

 

[nns91]

Thanks. Can you explain more on how should I find the unit vector ?

 

[Dr.D]

In plane polar coordinates, the unit vectors er and etheta play the same sort of roles that i and j play in cartesian coordinates, although they are not constants. So, we might say that they are simply defined. They are what are called "basis vectors" for the space.

Have you not talked about them in your class? This is tough stuff to be working if you don't have the tools!

 

[nns91]

Yeah, I know that v= r ur + rtheta utheta.

However, in this particular case, how do I find r ur ?

 

[HallsofIvy]

Homework Statement

A slender rod through the origin of the polar plane rotates about the origin at the rate of 3 rad/min. A beetle starting from the point (2,) crawls along the rod toward the origin at the rate of 1 in/min. The axes are marked in inches

a. Find the beetle's acceleration and velocity in polar form when it is halfway to (1 inch from) the origin.
b.To the nearest thousandth of an inch, what will be the length of the path the beetle has traveled by the time it reaches the origin.

Homework Equations

acceleration, velocity, arc length

The Attempt at a Solution

Can anyone hint me how to do this problem ?

For part b, I guess I just take the integral of speed from 0 to t in which t is the time the beetle needs to reach the origin ??

The beetles distance from the origin, after t min. is r(t)= 2- 2t. Setting up a coordinate system with x-axis along the initial position of the rod, $\theta$, the angle the rod makes with the x-axis, after t min, is given, in radians, by $\theta= 3t$. Since $(x,y)= (r cos(\theta), r sin(\theta))$, then the position of the beetle is given by $(2(1-t)cos(3t), 2(1-t)sin(3t))$.

Find the velocity and acceleration vectors by differentiating that. The arclength is given by
$$\int \sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dt}\right)^2}dt$$
where the integration is from t= 0 to the t at which the beetle reaches the origin (which is easy).