Exercise 5 Find the center of mass of the uniformly dense upper half disk x² + y² ≤ R², y 20. Your answer may depend on R. Exercise 6 Find the moment of inertia around the origin of the uniformly dense upper half disk x² + y² ≤ R², y 20 of total mass M. Your answer may depend on R and M.
Exercise 5 Find the center of mass of the uniformly dense upper half disk x² + y² ≤ R², y 20. Your answer may depend on R. Exercise 6 Find the moment of inertia around the origin of the uniformly dense upper half disk x² + y² ≤ R², y 20 of total mass M. Your answer may depend on R and M.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please solve both 5 and 6 will be thumbs up
Transcribed Image Text:Exercise 5 Find the center of mass of the uniformly dense upper half disk
x² + y² ≤ R², y 20. Your answer may depend on R.
Exercise 6 Find the moment of inertia around the origin of the uniformly
dense upper half disk x² + y² ≤ R2, y ≥ 0 of total mass M. Your answer
may depend on R and M.
![These exercises serve as homework for our lesson on probability/variance and
centers of mass/moments of inertia. In what follows, fx(x) is the probability
density function of a random variable X and 6(x) is the mass density function
of some object, where x ER". Also, we will use the shorthand f f = fRn f(x) dx.
Here are some formulas we discussed in class. You may not need all of them,
I'm only providing them for reference.
Probability
Mean/Expected Value
P(XEA) = fx(x)
E[X]=[xfx(x)
Covariance
Marginal Density
Cov(X,Y)= E(X - E[X])(Y - E[Y])]
fx = ffx,y(x, y) dy
Variance
Var(X) = Cov(X, X) = E[(X - E[X])²]
Standard Deviation
o(X)=√Var(X)
Ej = Cov(X₁, X₂)
Covariance Matrix
Mass
M = 8(x)
Jx8(x)
Center of Mass
M
Moment of Inertia I= [d(x)²8(x)
Also recall that something is uniformly distributed over a set A if its den-
sity function is a constant in A and zero outside A. Thus, if X is uniformly
distributed in A and B C A, then P(XB) = (B), where measures the
length, area, volume, etc.
H(A)'](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F444a28ed-437f-4929-a1ec-5907998afa93%2F219f46df-942e-4598-8b04-85f0b3cb772e%2Fwi59a8s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:These exercises serve as homework for our lesson on probability/variance and
centers of mass/moments of inertia. In what follows, fx(x) is the probability
density function of a random variable X and 6(x) is the mass density function
of some object, where x ER". Also, we will use the shorthand f f = fRn f(x) dx.
Here are some formulas we discussed in class. You may not need all of them,
I'm only providing them for reference.
Probability
Mean/Expected Value
P(XEA) = fx(x)
E[X]=[xfx(x)
Covariance
Marginal Density
Cov(X,Y)= E(X - E[X])(Y - E[Y])]
fx = ffx,y(x, y) dy
Variance
Var(X) = Cov(X, X) = E[(X - E[X])²]
Standard Deviation
o(X)=√Var(X)
Ej = Cov(X₁, X₂)
Covariance Matrix
Mass
M = 8(x)
Jx8(x)
Center of Mass
M
Moment of Inertia I= [d(x)²8(x)
Also recall that something is uniformly distributed over a set A if its den-
sity function is a constant in A and zero outside A. Thus, if X is uniformly
distributed in A and B C A, then P(XB) = (B), where measures the
length, area, volume, etc.
H(A)'
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