Vector Problem: Acceleration, Velocity & Arc Length

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In summary: So, the acceleration vector is given by a=\frac{\left(\frac{2}{3}\right)^3}{2}=1.5\frac{\mathrm{rad}/\mathrm{sec}}{\mathrm{min}}, and the velocity vector is given by v=\frac{\left(\frac{2}{3}\right)^3}{2}=2\frac{\mathrm{rad}/\mathrm{sec}}{\mathrm{min}}, both in radians per second.
  • #1
nns91
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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 ??
 
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  • #2
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.
 
  • #3
Thanks. Can you explain more on how should I find the unit vector ?
 
  • #4
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!
 
  • #5
Yeah, I know that v= r ur + rtheta utheta.

However, in this particular case, how do I find r ur ?
 
  • #6
nns91 said:

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, [itex]\theta[/itex], the angle the rod makes with the x-axis, after t min, is given, in radians, by [itex]\theta= 3t[/itex]. Since [itex](x,y)= (r cos(\theta), r sin(\theta))[/itex], then the position of the beetle is given by [itex](2(1-t)cos(3t), 2(1-t)sin(3t))[/itex].

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

1. What is the difference between acceleration and velocity?

Acceleration is the rate of change of velocity over time. It measures how quickly the velocity of an object is changing. Velocity, on the other hand, is the rate of change of an object's position over time. It measures the speed and direction of an object's motion.

2. How do you calculate acceleration?

Acceleration can be calculated by dividing the change in velocity by the change in time. The formula for acceleration is: a = (vf - vi) / t, where a is acceleration, vf is final velocity, vi is initial velocity, and t is time.

3. What is arc length and how is it related to velocity?

Arc length is the distance along a curve or arc. In the context of velocity, arc length is the distance traveled by an object over a curved path. It is related to velocity because the longer the arc length, the longer the distance traveled, and thus, the higher the velocity of the object.

4. What is the relationship between acceleration and velocity?

Acceleration and velocity are related in that acceleration is the rate of change of velocity. This means that when an object is accelerating, its velocity is changing. A positive acceleration means the object's velocity is increasing, while a negative acceleration means it is decreasing.

5. How is the vector problem of acceleration, velocity, and arc length used in real life?

The vector problem of acceleration, velocity, and arc length is used in various real-life situations, such as calculating the speed of a car, the trajectory of a projectile, or the motion of a rollercoaster. It is also essential in fields like physics, engineering, and sports, where understanding and manipulating velocity and acceleration are crucial for success.

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