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Answer :
The total kinetic energy of the system is 481 J, thus none of the given option is correct and the correct answer is 481 J.
To find the total kinetic energy of the system, we need to calculate the kinetic energy of the merry-go-round and the kinetic energy of the child separately and then add them together.
First, let's find the kinetic energy of the merry-go-round. The formula for the kinetic energy of a rotating object is:
Kinetic Energy (KE) = 0.5 * I * ω²
where I is the moment of inertia and ω is the angular velocity.
For a solid disk, the moment of inertia (I) is given by:
I = (1/2) * M * R²
where M is the mass and R is the radius of the merry-go-round.
Plugging in the given values:
I = (1/2) * 50.0 kg * (1.50 m)²
I = 187.5 kg*m²
Now, we need to convert the angular velocity (20.0 rpm) to radians per second (rad/s):
1 rpm = 2π rad/min
20.0 rpm = 20.0 * 2π rad/min = 40π rad/s
Now we can calculate the kinetic energy of the merry-go-round:
KE[tex]_{merry_{go_{round[/tex] = 0.5 * 187.5 kg*m² * (40π rad/s)²
KE[tex]_{merry_{go_{round[/tex] ≈ 185 J
Now let's find the kinetic energy of the child. The formula for the kinetic energy of an object is:
KE = 0.5 * m * v²
where m is the mass and v is the velocity.
First, we need to find the velocity of the child. We can use the formula:
v = ω * R
where ω is the angular velocity and R is the distance from the center.
The angular velocity of the child is the same as the merry-go-round, which is 40π rad/s. The distance from the center of the child is 1.25 m.
v = 40π rad/s * 1.25 m
v ≈ 101.25 m/s
Now we can calculate the kinetic energy of the child:
KE[tex]_{child[/tex] = 0.5 * 18.0 kg * (101.25 m/s)²
KE[tex]_{child[/tex] ≈ 296 J
Finally, we add the kinetic energies of the merry-go-round and the child to find the total kinetic energy of the system:
Total KE = KE[tex]_{merry_{go_{round[/tex] + KE[tex]_{child[/tex]
Total KE = 185 J + 296 J
Total KE ≈ 481 J
Thus none of the given option is correct and the correct answer is 481 J.
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