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Ivan
3 years ago
10

A solid, horizontal cylinder of mass 10.6 kg and radius 1.00 m rotates with an angular speed of 8.00 rad/s about a fixed vertica

l axis through its center. A 0.250-kg piece of putty is dropped vertically onto the cylinder at a point 0.900 m from the center of rotation and sticks to the cylinder. Determine the final angular speed of the system.
Physics
1 answer:
n200080 [17]3 years ago
5 0

Answer:

The final angular speed is 7.71 rad/s

Explanation:

Given

Cylinder mass, M = 10.6 kg

Cylinder radius, R = 1.00 m

Angular speed, w = 8.00 rad/s.

Mass of putty, m = 0.250-kg

Radius, r = 0.900 m

First, we set up an expression for the initial and final angular momentum of the system.

The moment of inertia of the cylinder is given as I = ½MR²

While the moment of inertia if the putty is mr².

Initial Momentum of the system = Initial momentum of the cylinder =

Li = Iw --- Substitute ½MR² for I

Li = ½MR²w

By

Substituton

Li = ½ * 10.6 * 1² * 8

Li = 42.4kgm²/s

Calculating the final momentum of the system.

First we calculate the final momentum of the cylinder

Li = Iw --- Substitute ½MR² for I

Li = ½MR²wf where wf = final angular speed

By

Substituton

Li = ½ * 10.6 * 1² wf

Li = 5.3w kgm²/s

Then we calculate the final momentum of the putty

Final Momentum of the putty =

L2 = Iwf --- Substitute mr² for I;

L2 = mr²wf --- By Substituton

L2 = 0.25 * 0.9² * wf

L2 = 0.2025wf kgm²/s

Final momentum = Li + L2

Lf = (5.3wf + 0.2025wf) kgm²/s

Lf = 5.5025wf kgm²/s

By conservation of momentum

Li = Lf

Where Li = 42.4kgm²/s and Lf = 5.5025wf kgm²/s

So, we have

5.5025wf kgm²/s = 42.4kgm²/s --- make wf the subject of formula

wf = 42.4/5.5025

wf = 7.71 rad/s

Hence, the final angular speed is 7.71 rad/s

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          A₁ v₁ = A₂ v₂

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          r₁ = d₁ / 2 = 2.25 / 2 = 1,125 cm

          r₂ = d₂ / 2 = 0.2 / 2 = 0.100 cm

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          A₁ = 3,976 cm²

          A₂ = π r₂²

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Now with the continuity equation we can look for the speed of water inside the hose

           v₁ = v₂ A₂ / A₁

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Now we can use Bernoulli's equation, pa pressure at the nozzle is the air pressure (P₂ = Patm) the hose must be on the floor so the height is zero (y₁ = 0)

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          P₁ = Patm + ½ ρ (v₂² - v₁²) + ρ g y₂

Let's calculate

           P₁ = 1.013 10⁵ + ½ 1000 (11.2² - 0.1273²) + 1000 9.8 7.25

           P₁ = 1.013 10⁵ + 6.271 10⁴ + 7.105 10⁴

           P₁ = 2.3506 10⁵ Pa

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