Answer:
a)
b)
Explanation:
From the question we are told that:
Cast-iron block Dimension:
Length
Width
Feed
Depth
Diameter
Number of cutting teeth
Rotation speed
Generally the equation for Approach is mathematically given by
Therefore
Effective length is given as
a)
Generally the equation for Machine Time is mathematically given by
Where
Therefore
b)
Generally the equation for Material Removal Rate. is mathematically given by
Let <em>f(z)</em> = (4<em>z </em>² + 2<em>z</em>) / (2<em>z </em>² - 3<em>z</em> + 1).
First, carry out the division:
<em>f(z)</em> = 2 + (8<em>z</em> - 2) / (2<em>z </em>² - 3<em>z</em> + 1)
Observe that
2<em>z </em>² - 3<em>z</em> + 1 = (2<em>z</em> - 1) (<em>z</em> - 1)
so you can separate the rational part of <em>f(z)</em> into partial fractions. We have
(8<em>z</em> - 2) / (2<em>z </em>² - 3<em>z</em> + 1) = <em>a</em> / (2<em>z</em> - 1) + <em>b</em> / (<em>z</em> - 1)
8<em>z</em> - 2 = <em>a</em> (<em>z</em> - 1) + <em>b</em> (2<em>z</em> - 1)
8<em>z</em> - 2 = (<em>a</em> + 2<em>b</em>) <em>z</em> - (<em>a</em> + <em>b</em>)
so that <em>a</em> + 2<em>b</em> = 8 and <em>a</em> + <em>b</em> = 2, yielding <em>a</em> = -4 and <em>b</em> = 6.
So we have
<em>f(z)</em> = 2 - 4 / (2<em>z</em> - 1) + 6 / (<em>z</em> - 1)
or
<em>f(z)</em> = 2 - (2/<em>z</em>) (1 / (1 - 1/(2<em>z</em>))) + (6/<em>z</em>) (1 / (1 - 1/<em>z</em>))
Recall that for |<em>z</em>| < 1, we have
Replace <em>z</em> with 1/<em>z</em> to get
so that by substitution, we can write
Now condense <em>f(z)</em> into one series:
So, the inverse <em>Z</em> transform of <em>f(z)</em> is .
Answer:
Explanation:
Previous concepts
Angular momentum. If we consider a particle of mass m, with velocity v, moving under the influence of a force F. The angular momentum about point O is defined as the “moment” of the particle’s linear momentum, L, about O. And the correct formula is:
Applying Newton’s second law to the right hand side of the above equation, we have that r ×ma = r ×F =
MO, where MO is the moment of the force F about point O. The equation expressing the rate of change of angular momentum is this one:
MO = H˙ O
Principle of Angular Impulse and Momentum
The equation MO = H˙ O gives us the instantaneous relation between the moment and the time rate of change of angular momentum. Imagine now that the force considered acts on a particle between time t1 and time t2. The equation MO = H˙ O can then be integrated in time to obtain this:
Solution to the problem
For this case we can use the principle of angular impulse and momentum that states "The mass moment of inertia of a gear about its mass center is ".
If we analyze the staritning point we see that the initial velocity can be founded like this:
And if we look the figure attached we can use the point A as a reference to calculate the angular impulse and momentum equation, like this:
And if we integrate the left part and we simplify the right part we have
And if we solve for we got:
