Two astronaut, as shown in the figure, each having a mass of 62.0 kg, are connected by a 12.00 m rope of negligible mass. They are isolated in space, moving in circles around the point halfway between them at a speed of 5.00 m/s. Treating the astronauts as particles, calculate (b) the rotational energy of the system.
The rotational energy of each astronaut is
1/2 * I * omega^2
where I is the moment of inertia, and omega is the angular speed.
The moment inertia of the astronaut is m*r^2
r = 12.0 / 2 = 6 m
v = 5 m/s = omega*r
The total rotational energy is
1/2 * I * omega^2 + 1/2 * I * omega^2 = I * omega^2
To calculate the rotational energy of the system, we need to use the formula for the rotational kinetic energy:
Rotational kinetic energy (Krot) = 0.5 * moment of inertia (I) * angular velocity (ω)^2
In this case, the two astronauts are moving in circles around the point halfway between them, which means they are rotating about an axis. Since the problem states that we should treat the astronauts as particles, we assume that they have a moment of inertia equivalent to that of a point mass.
The moment of inertia for a point mass rotating about an axis is given by the formula:
I = m * r^2
Where:
- I is the moment of inertia
- m is the mass of the point mass
- r is the distance between the point mass and the axis of rotation
In this case, the distance between each astronaut and the axis of rotation is half of the 12.00 m rope, so r = 12.00 m / 2 = 6.00 m.
Now, we can substitute the values into the formulas to calculate the rotational energy:
Rotational kinetic energy (Krot) = 0.5 * I * ω^2
Since the astronauts are moving in circles at a constant speed of 5.00 m/s, we can calculate the angular velocity (ω) using the formula:
ω = v / r
Where:
- ω is the angular velocity
- v is the linear velocity of the astronauts
- r is the distance between the astronauts and the axis of rotation
Plugging in the values, we get:
ω = 5.00 m/s / 6.00 m
Now we can calculate ω and substitute the values into the rotational kinetic energy formula to find the answer.