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26 Nov 2019

I have to prove that F=Fo((1-T)/T)breaks down to v(T)=vo+ao(T/2) and x(T)=voT+ao((T2)/3). I amcontinuously finding flaws in my integrals from acceleration tovelocity (and even going from velocity to position, and both aregiven!). I know that I can get a(T) bydividing F(T) by mass "m"., but from there, nothing linesup.

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Related textbook solutions

Physics: Principles with Applications
7th Edition, 2013
Giancoli
ISBN: 9780321625922

Related questions

A mass m rests on a frictionless horizontal table and isconnected to rigid supports via 2 identical springs each of relaxedlength L0 and spring constantk. Each spring is stretched to a length L considerable greater thanL0. Horizontal displacementsof m from its equilibrium position are labeled x and y.
(Diagram insert didn't work, all it really tells you is that xis on the axis of the springs and y is perpendicular to thesprings)
a) Write down the differential equation of motion (i.e.,Newton's Law) governing small oscillations in the xdirection.
b) Same as a but in the y direction (assume that y <<L).
c) In terms of L and L0, calculate the ratio of the periods ofthe oscillations along x and y.
d) If at t=0 the mass m is released from the point x = y = A0with zero velocity, what are its x and y coordinates at any latertime t?
---------------------------
So, I started with just ma = -2kx for a, leading me to thedifferential equation x''(t) + (2k/m) x = 0.
Then I did basically the same thing with y.
However, when I get to c, there's my problem - I don't knowhow I could write any of this in L and L0.
Also, b doesn't really make sense to me because if y is smallwhen compared to L, it seems like there shouldn't be anyoscillation in the y direction.
The answer the back of the book shows for c and d are:
c) Tx/Ty = (1 - L0 / L)1/2
d) x(t) = A0cos( (2k/m)1/2 t)
y(t) = A0cos( (2k(L-L0)/mL)1/2 t)
I have no idea how they got to these answers. Can somebodyhelp me? It has something to do with springs already beingstretched, but it seems they are making an assumption somewherethat I don't understand.

Question: An automobile traveling 30 mi/hr has wheels 28 inchesin diameter. If the car is brought to a stop uniformly in 78 turnsof the wheels, what is the magnitude of the angular acceleration ofthe wheels (in radians/second^2)?

As far as the concept goes, I am looking for the angularacceleration, so what I did to make my way toward that was to setvelocity of center mass (automobile's speed) equal to radius *angular velocity:
vcm = rω

and I solved for ω which gives me the value of angular velocity(note, that I took units into consideration). I do not know whereto go from there. I tried to plug in the values for the periodformula T=2πr/v but I do not know if I'm on the right track.

Would highly appreciate a walkthrough. Thanks!

1. An interesting one-dimensional system is the simple pendulum,consisting of a point mass m, fixed to the end of a massless rod(length l), whose other end is pivoted from the ceiling to let itswing freely in a vertical plane, as shown in figure below. Thependulum’s position can be specified by its angle ? from theequilibrium position. (It could be equally specified by itsdistance s from equilibrium, i.e. s = l?, but the angle is moreconvenient).
(a) Prove that the pendulum’s potential energy (measured from theequilibrium level) is


(b) Write down the total energy E as a function of ? and .
(c) Show that by differentiating your expression for E with respectto t you can get the equation of motion for ? and that the equationof motion is just the familiar (where is the torque, I is themoment of inertia, and ? is the angular acceleration)
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