Two astronauts (Fig. P8.68), each having a mass of 84.0 kg, are connected by a 10.0 m rope of negligible mass. They are isolated in space, moving in circles around the point halfway betwen them at a speed of 5.80 m/s. Treating the astronauts as particles, calculate each of the following.
(a) the magnitude of the angular momentum of the system
kg·m2/s
(b) the rotational energy of the system
kJ
By pulling on the rope, the astronauts shorten the distance between them to 5.00 m.
(c) What is the new angular momentum of the system?
kg·m2/s
(d) What are their new speeds?
m/s
(e) What is the new rotational energy of the system?
kJ
(f) How much work is done by the astronauts in shortening the rope?
kJ
(Could someone tell me how to do each part?)
(a) To calculate the magnitude of angular momentum, we need to use the formula:
Angular Momentum (L) = Moment of Inertia (I) x Angular Velocity (ω)
Since the astronauts are treated as particles rotating in circles, we can assume that they can be represented by point masses. The moment of inertia for a point mass rotating around a fixed point is given by:
Moment of Inertia (I) = mass (m) x radius of rotation (r)²
In this case, both astronauts have the same mass (84.0 kg) and are moving in circles with a radius of 10.0 m. Thus, we can calculate the moment of inertia for each astronaut:
Moment of Inertia (I) = (mass of one astronaut) x (radius of rotation)²
I = (84.0 kg) x (10.0 m)²
Now, we need to calculate the angular velocity (ω). The linear velocity (v) can be found using the equation:
v = ω x r
Given that the astronauts are moving with a speed of 5.80 m/s and the radius of rotation is 10.0 m, we can solve for ω:
5.8 m/s = ω x 10.0 m
ω = 5.8 m/s ÷ 10.0 m
Now, we have all the information needed to calculate the angular momentum:
L = I x ω
(b) The rotational energy of the system is given by the equation:
Rotational Energy (E) = 1/2 x I x ω²
Now that we have I and ω from part (a), we can substitute them into this formula to find the rotational energy of the system.
(c) When the astronauts shorten the rope to 5.00 m, the radius of rotation changes. We need to calculate the new moment of inertia (I) using the new radius, and then find the new angular momentum (L) using the same formula as in part (a).
(d) To find the new speeds of the astronauts, we can use the relationship between linear velocity (v) and angular velocity (ω) as we did in part (a):
v = ω x r
Now, the radius of rotation is 5.00 m, so we can solve for ω and then find the new speeds of the astronauts.
(e) Similar to part (b), we can calculate the new rotational energy of the system using the new moment of inertia (I) and the new angular velocity (ω) obtained in part (d).
(f) To determine the work done by the astronauts in shortening the rope, we need to calculate the change in potential energy. The initial potential energy of the system is given by:
Initial Potential Energy = mass (m) x gravitational acceleration (g) x initial distance between the astronauts
The final potential energy can be calculated using the same formula, but with the final distance between the astronauts. The work done will be the change in potential energy.