Two ice skaters hold hands and rotate, making one revolution in 7.6 s. Their masses are 15 kg and 50 kg, and they are separated by 4.6 m.

Find the angular momentum of the system
about their center of mass.
Answer in units of J · s
Find the total kinetic energy of the system.
Answer in units of J

Distance between the skaters is d, frequency is n =1/7.6 s^-1

Location of the center of mass
of the second skater of mass m2 is
x1 = d •m2/(m1+m2),
x2 = d - x1,
Moment of inertia of the system is
I= I1 +I2 =
= m1•x1^2 + m2•x2^2.
Angular momentum is
L = I•ω= I•2•π•n.
KE = I• ω^2/2 = I•(2•π•n)^2/2.
Check your given data. It seems to me that the distance 4.6 m is too much if the “skaters hold hands”. Moreover, the mass m1 = 15 kg is improbable for skater (this is the mass of the child of three).

To find the angular momentum of the system about their center of mass, we can use the formula:

Angular momentum = Moment of inertia * Angular velocity

The moment of inertia of a system can be calculated as the sum of the individual moments of inertia, given by the equation:

Moment of inertia = mass * radius^2

where mass is the mass of each skater and radius is the distance of each skater from the center of mass of the system.

First, let's find the moment of inertia of each skater about the center of mass:

For the skater with a mass of 15 kg:
Moment of inertia1 = 15 kg * (4.6 m/2)^2 = 15 kg * 10.65 m^2 = 159.75 kg m^2

For the skater with a mass of 50 kg:
Moment of inertia2 = 50 kg * (4.6 m/2)^2 = 50 kg * 10.65 m^2 = 532.5 kg m^2

Next, let's find the total moment of inertia of the system about their center of mass:

Total moment of inertia = Moment of inertia1 + Moment of inertia2 = 159.75 kg m^2 + 532.5 kg m^2 = 692.25 kg m^2

Now, let's find the angular velocity of the system. Since they make one revolution in 7.6 s, we can calculate the angular velocity using the formula:

Angular velocity = (2π rad)/(time taken)

Angular velocity = (2π rad)/(7.6 s) ≈ 0.822 rad/s (rounded to three decimal places)

Finally, we can find the angular momentum using the formula:

Angular momentum = Total moment of inertia * Angular velocity

Angular momentum = 692.25 kg m^2 * 0.822 rad/s = 569.23 J · s (rounded to two decimal places)

Therefore, the angular momentum of the system about their center of mass is approximately 569.23 J · s.

Moving on to the total kinetic energy of the system, it can be calculated as the sum of the individual kinetic energies of each skater:

Kinetic energy1 = (1/2) * mass1 * velocity1^2

where mass1 is the mass of the first skater and velocity1 is the velocity of the first skater.

For the skater with a mass of 15 kg, we need to find the velocity. Since they are rotating together, the velocity can be calculated as the product of the angular velocity and the radius:

velocity1 = angular velocity * (4.6 m/2) = 0.822 rad/s * 2.3 m = 1.89 m/s (rounded to two decimal places)

Now, let's calculate the kinetic energy for skater 1:

Kinetic energy1 = (1/2) * 15 kg * (1.89 m/s)^2 = 20.18 J (rounded to two decimal places)

Similarly, we can calculate the kinetic energy for skater 2:

velocity2 = angular velocity * (4.6 m/2) = 0.822 rad/s * 2.3 m = 1.89 m/s (rounded to two decimal places)

Kinetic energy2 = (1/2) * 50 kg * (1.89 m/s)^2 = 89.74 J (rounded to two decimal places)

Finally, we can find the total kinetic energy of the system by adding the individual kinetic energies:

Total kinetic energy = Kinetic energy1 + Kinetic energy2 = 20.18 J + 89.74 J = 109.92 J (rounded to two decimal places)

Therefore, the total kinetic energy of the system is approximately 109.92 J.