What is the new speed v’ away from the wall when rolling motion is re-established?

  1. A wheel (mass M, radius R) rolls with speed v and angular velocity w (v=?R) and collides elastically with a wall. Immediately after the collision it has v away from the wall, but the same ? as before – so it’s skidding, not rolling.
    Friction at the floor will eventually re-establish rolling motion, by doing negative work on the system and applying a torque that will change ?.
    What is the new speed v’ away from the wall when rolling motion is re-established?

    Answer by RickB
    That’s an interesting question!

    First, let’s set up direction conventions. Let’s say:

    * The wall is on the left, so the wheel bounces off to the right. Let’s call “rightward” the positive linear direction.
    * Let’s call “clockwise” the positive angular direction. (That means the wheel’s rotation is negative immediately after the bounce.)

    During skidding, the force of friction is more or less constant (given by: (normal force)×(coef. of sliding friction)), so we can say that “Ff” is the (constant) force of friction during the skidding phase.

    The force Ff produces a torque on the wheel, in the amount of: ? = (Ff)R. (This is a positive torque, since Ff points to the left and therefore tends to drive the wheel clockwise.) We know that the angular acceleration is given by:

    ? = ?/I = (Ff)R/I

    where “I” is the wheel’s moment of inertia. I’m not sure what kind of wheel this is, but its moment of inertial can be represented as: I = ?mR², for some “?” that depends on how the wheel’s mass is distributed. (It’s some constant between 0 and 1.) So we can write:

    ? = (Ff)/(?mR)

    At the same time, the ball’s linear acceleration is:

    a = -Ff/m

    (Here we use negative Ff, since the force points toward the left)

    With these two values, we can write the ball’s angular and linear velocities as a function of time:

    ?(t) = ?0 + ?t = ?0 + (Ff)t/(?mR)
    (where ?0 is the ball’s angular velocity right after collision)

    and:
    v(t) = v0 + at = v0 + (?Ff/m)t
    (where v0 is the ball’s linear velocity right after collision)

    We are also given that ?0 = ?v0/R, so we can rewrite the ?(t) equation as:

    ?(t) = ?v0/R + (Ff)t/(?mR)

    Now, at some time t?, rolling is re-established. That means:

    v(t?) = ?(t?)R
    or:
    v0 + (-Ff/m)t? = ?v0 + (Ff)t?/(?m)

    From which:
    t? = 2(v0)m?/[Ff(1+?)]

    Now plug that value back into the “v(t)” equation:

    v? = v(t?) = v0 + (-Ff/m)2(v0)m?/[Ff(1+?)]
    = v0 – 2(v0)?/(1+?)
    = v0(1 – 2?/(1+?))
    = v0(1-?)/(1+?)

    So for example, if you assume that the wheel has a uniform density, then ?=½, in which case v? = v0/3.

  2. I own 1 unisex deep v and 1 unisex regular v-neck. I know that the regular v-neck runs slightly bigger than the deep, but how do they compare to the regular t-shirts as far as fit? I wear an xs in both v-necks for reference.

    Answer by Benjamin
    The fit of most of the crew necks (2001, BB401, TR401, 6401) is closer to the regular v neck (2456)

    Go with XS in unisex styles and a S in womens.

  3. What current would flow in an ordinary 110 V household circuit, if a 1210 W hair drier and a 700 W microwave oven were operating simultaneously on this line?
    Answer – 17.36 A

    How much current would the hair drier and microwave from the previous question draw, if they were connected to the 110 V line in series?

    Answer by Kim
    IF you assume they act like simple resistors, then you can figure out their individual resistances using P = I^2 R. In the first part, you know P and I.

    Once you know their resistances, when you put them in series you add their resistances. So then you can find I from I = V/R.

    But I’m not sure these devices will really act like pure resistors because they have motors in them. There could be an inductive component.

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First, I will get in contact with my local political representatives. I will inform them on the severity of this situation. They have the power to make and shape political policies to solve this problem. Politicians can inform other political figures and spread awareness to them. I will vote for politicians that care deeply about this topic and are willing to solve this problem. Politicians are necessary players in the journey for affordable internet access.

Next, I will contact internet companies. I will talk to the leaders of these corporations. I will plead my case to them in a polite and professional manner. I will tell them of the importance of having affordable internet access. I will tell them of the benefits of this movement. People who have affordable internet will be able to contribute to society. They can use the internet to study and research. Those same young people will grow up and possibly might work for their companies in the future. Their contributions of these companies would be in their best interest. Their affordable internet access will shape future leaders of the world.

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