Homework H1.E - Fa22

Problem statement
Solution video

DISCUSSION THREAD

Discusssion

As you watch the sphere move through its path, focus on a few things:

  • Direction of the velocity vector. The velocity of the ball is always tangent to the path, as expected. Note that the unit tangent vector e_t points in the same direction as the velocity.
  • Direction of the acceleration vector. The acceleration of the ball always points "inward" on the path, again, as expected. When P is slowing down, you see that the acceleration vector points in the negative e_t direction. Similarly, when P is increasing in speed, the acceleration vector points in the positive e_t direction.
  • Directions of the polar unit vectors. The e_r unit vector always points outward from  point O toward the sphere. The e_θ unit vector is perpendicular to e_r, and in the direction of increasing θ, as seen in the animation above.

Your task is to calculate R, R_dot and R_ddot using the given equation relating R and θ. This will require derivatives with respect to TIME; however, since R is given in terms of θ, you will need to use the chain rule of differentiation. For example: R_dot = dR/dt = (dR/dθ)(dθ/dt).


Any questions?? Please ask/answer questions regarding this homework problem through the "Leave a Comment" link above.

9 thoughts on “Homework H1.E - Fa22”

    1. So in the image given in the hw, there is an arrow indicating the direction of theta. I think you can also say that this arrow shows the direction of motion. This is the positive theta direction (similar to say, the positive x direction) and the direction of e_theta always points in the same direction. Does that make any sense?

    2. What Katherine says here is correct. The direction indicated by the arrow on theta is the direction of motion when theta is increasing (or positive). It may or may not be the actual direction of motion, just the direction when theta is defined as positive. Just like if you draw the x-direction to the right - x is positive then when the motion is to the right. And, in contrast, x would be negative for motion to the left.

  1. In the animation, the arrows representing velocity and acceleration seem to not always align with e^theta and e^r. How do we determine their directions if they are just magnitudes of the two e's in the velocity and acceleration equation?

    1. ENRICO: Not sure of your question here. The e_R and e_theta components of velocity are found directly from the time derivatives of R and theta. Similarly, for the e_R and e_theta components of acceleration. Calculate these for the given position of theta = pi/2, and you have the answers.

      Your observation is correct in that the velocity and acceleration vectors are almost never aligned with either e_R or e_theta.

      1. I think I understand now, it just happens that at theta = pi/2 the velocity only has an e_theta component and acceleration only has an e_r component, making them aligned with the respective e vectors. But at other angles such as pi/4, the velocity and acceleration would both have an e_theta and e_r component, so the sum of both those vectors would mean the velocity and acceleration no longer align with e_theta and e_r.

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