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densk [106]
3 years ago
5

A playground merry-go-round of radius R = 1.80 m has a moment of inertia I = 255 kg · m2 and is rotating at 9.0 rev/min about a

frictionless vertical axle. Facing the axle, a 24.0-kg child hops onto the merry-go-round and manages to sit down on the edge. What is the new angular speed of the merry-go-round?
Mathematics
1 answer:
Degger [83]3 years ago
3 0

Answer:

7 rpm = 0.73 rad/s

Step-by-step explanation:

R = Radius of merry-go-round = 1.8 m

I_M= Moment of inertia of merry-go-round = 255 kg m²

I_C= Moment of inertia of child

ω = 9 rev/min

m = Mass of child = 24 kg

From the conservation of angular momentum

I\omega=I'\omega '\\\Rightarrow I\omega=(I_M+I_C)\omega'\\\Rightarrow \omega'= \frac{I\omega}{(I_M+I_C)}\\\Rightarrow \omega'=\frac{I\omega}{(I_M+mR^2)}\\\Rightarrow \omega'=\frac{255\times 9}{(255+24\times 1.8^2)}\\\Rightarrow \omega'=6.9\ rev/min

∴ New angular speed of the merry-go-round is 7 rpm = 7\times \frac{2\pi}{60}=\mathbf{0.73\ rad/s}

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Answer:

(a) l = -w + 21

(b) Domain: 0 <em>(See attachment for graph)</em>

(c) f(w) = -w + 21

Step-by-step explanation:

Given

2(l + w) = 42

l = length

w = width

Solving (a): A function; l in terms of w

All we need to do is make l the subject in 2(l + w) = 42

Divide through by 2

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Subtract w from both sides

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Reorder

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Solving (b): The graph

In (a), we have:

l = -w + 21

Since l and w are the dimensions of the fence, they can't be less than 1

So, the domain of the function can be 0

--------------------------------------------------------------------------------------------------

To check this

When w = 1

l = -1 + 21

l = 20

(w,l) = (1,20)

When w = 20

l = -20 + 21

l = 1

(w,l)= (20,1)

--------------------------------------------------------------------------------------------------

<em>See attachment for graph</em>

<em></em>

Solving (c): Write l as a function f(w)

In (a), we have:

l = -w + 21

Writing l as a function, we have:

l = f(w)

Substitute f(w) for l in l = -w + 21

l = -w + 21 becomes

f(w) = -w + 21

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