Two astronauts, each having a mass of 78.5 kg, are connected by a 10.0-m rope of negligible mass. They are isolated in space, moving in circles around the point halfway between them at a speed of 5.50 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
Hints:
when they shorten the rope, there is no external moment on he system so the angular momentum CAN NOT change. Get the new (higher)speeds from conservation of angular momentum.
they did work shortening the rope so the energy goes up
the work done equals the change in kinetic energy of the system
To calculate each of the quantities, we can use the following formulas:
(a) Angular momentum (L) = moment of inertia (I) x angular velocity (ω)
(b) Rotational energy (E) = 0.5 x moment of inertia (I) x angular velocity (ω)^2
Given information:
Mass of each astronaut (m) = 78.5 kg
Distance between the astronauts (r) = 10.0 m
Initial velocity (v) = 5.50 m/s
Revised distance between the astronauts (r') = 5.00 m
(a) To calculate the magnitude of the angular momentum of the system:
The moment of inertia of the system can be calculated as follows:
Moment of inertia (I) = 2 * (m * r^2) / 2 = m * r^2
Angular velocity (ω) = v / r
Angular momentum (L) = I * ω = (m * r^2) * (v / r)
Substituting the values:
L = (78.5 kg * (10.0 m)^2) * (5.50 m/s / 10.0 m)
(b) To calculate the rotational energy of the system:
Rotational energy (E) = 0.5 * I * (ω^2) = 0.5 * (m * r^2) * (v / r)^2
Substituting the values:
E = 0.5 * (78.5 kg * (10.0 m)^2) * (5.50 m/s / 10.0 m)^2
(c) To calculate the new angular momentum of the system:
The moment of inertia remains the same as the system hasn't changed shape, but the distance has been reduced.
So, the new moment of inertia (I') = m * r'^2
Angular momentum (L') = I' * ω = (m * r'^2) * (v / r)
(d) To calculate the new speeds:
To find the new speeds, we can use the principle of conservation of angular momentum:
Angular momentum before = Angular momentum after
(m * r^2) * (v / r) = (m * r'^2) * (v' / r')
(e) To calculate the new rotational energy of the system:
Rotational energy (E') = 0.5 * I' * (ω^2) = 0.5 * (m * r'^2) * (v' / r')^2
(f) To calculate the work done by the astronauts in shortening the rope:
Work done can be calculated as the change in the potential energy of the system.
Potential energy before = 0.5 * (m * r^2) * (v / r)^2
Potential energy after = 0.5 * (m * r'^2) * (v' / r')^2
Work done = Potential energy before - Potential energy after
To solve this problem, we need to use the principles of angular momentum, rotational energy, and work.
(a) To find the magnitude of the angular momentum of the system, we can use the formula:
Angular Momentum = Moment of Inertia * Angular Velocity
Since the astronauts are treated as particles, we can assume their moment of inertia is negligible. Therefore, the angular momentum of the system is simply the product of the mass of each astronaut, the distance between them, and their angular velocity:
Angular Momentum = (Mass1 + Mass2) * (Distance / 2) * Angular Velocity
Angular Momentum = (78.5 kg + 78.5 kg) * (10.0 m / 2) * 5.50 m/s
Calculating the above expression will give us the angular momentum in kg · m^2/s.
(b) To find the rotational energy of the system, we can use the formula:
Rotational Energy = 0.5 * Moment of Inertia * Angular Velocity^2
Since the moment of inertia is negligible, the rotational energy of the system is zero.
(c) When the astronauts shorten the distance between them to 5.00 m, their new angular momentum can be calculated using the same formula as in part (a):
New Angular Momentum = (Mass1 + Mass2) * (New Distance / 2) * Angular Velocity
New Angular Momentum = (78.5 kg + 78.5 kg) * (5.00 m / 2) * 5.50 m/s
Calculating the above expression will give us the new angular momentum in kg · m^2/s.
(d) To find the new speeds of the astronauts, we can use the fact that angular momentum is conserved in this system, which means the product of the moment of inertia and angular velocity remains constant. Therefore:
Initial Angular Momentum = New Angular Momentum
(Mass1 + Mass2) * (Distance / 2) * Initial Angular Velocity = (Mass1 + Mass2) * (New Distance / 2) * New Angular Velocity
Since the mass and angular velocity are the same for both astronauts, we can simplify the equation:
Distance * Initial Angular Velocity = New Distance * New Angular Velocity
Solving for New Angular Velocity:
New Angular Velocity = (Distance * Initial Angular Velocity) / New Distance
Using the given values, we can calculate the new angular velocity in m/s. The new speed of each astronaut will be the product of the new distance and the new angular velocity.
(e) Since the rotational energy of the system is zero irrespective of the distance between the astronauts, the new rotational energy will also be zero.
(f) The work done by the astronauts in shortening the rope can be calculated by finding the change in potential energy. The potential energy initially is zero, and the potential energy after shortening the rope is given by:
Potential Energy = (Mass1 * Mass2) / New Distance
Work Done = Change in Potential Energy = Potential Energy - Initial Potential Energy
Calculating the above expression will give us the work done by the astronauts in kJ.