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
These are pretty crazy measurements for human ice skaters but anyway
w = 2 pi/7.6 = .827 radians/second
find center of mass at x from big one
x(50)=(4.6-x)(15)
50 x = 69 - 15 x
65 x = 69
x = 1.06 m from big one and 3.54 from little one
I = 50(1.06)^2 + 15(3.54)^2
= 244 kg m^2
L = ang momentum = I w
= 244 (.827)
= 202 kg m^2/s
for KE
since you already have all the numbers...
KE1 + KE2 = (.5*I1*W1^2) + (.5*I2*W2^2)
Why did the ice skaters hold hands? Because they didn't want to drift apart and fall for each other!
To find the angular momentum of the system about their center of mass, we can use the formula:
L = Iω
where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
First, we need to find the moment of inertia of the system. The moment of inertia of a rotating object is given by the formula:
I = m*r^2
where m is the mass and r is the distance from the axis of rotation.
For the ice skater with a mass of 15 kg, the moment of inertia would be:
I1 = 15*(4.6/2)^2
And for the ice skater with a mass of 50 kg, the moment of inertia would be:
I2 = 50*(4.6/2)^2
Next, we need to find the angular velocity, which can be calculated by dividing the angle turned in radians by the time taken.
Since they make one revolution in 7.6 s, that means they go around the circle once, or 2π radians, in 7.6 seconds.
So, the angular velocity would be:
ω = (2π) / 7.6
Now we can plug these values into the formula for angular momentum:
L = I1*ω + I2*ω
Calculating this, we get the angular momentum of the system.
As for the total kinetic energy of the system, we can use the formula:
K.E. = 1/2 * I * ω^2
where I is the moment of inertia and ω is the angular velocity.
For the first ice skater, the kinetic energy would be:
K.E.1 = 1/2 * I1 * ω^2
And for the second ice skater, the kinetic energy would be:
K.E.2 = 1/2 * I2 * ω^2
The total kinetic energy of the system would then be:
Total K.E. = K.E.1 + K.E.2
Solving these equations will give you the values of the angular momentum and total kinetic energy in units of J · s and J respectively.
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
First, let's find the moment of inertia of the system. The moment of inertia of each skater can be calculated using the formula:
Moment of inertia = mass × radius^2
For the skater with a mass of 15 kg, the radius is half the separation distance between them, so the moment of inertia for this skater is:
Moment of inertia1 = 15 kg × (4.6 m/2)^2
For the skater with a mass of 50 kg, the radius is half the separation distance between them, so the moment of inertia for this skater is:
Moment of inertia2 = 50 kg × (4.6 m/2)^2
Now, let's calculate the angular velocity. Since they make one revolution in 7.6 seconds, the angular velocity can be found using the formula:
Angular velocity = 2π / time taken
Angular velocity = 2π / 7.6 s
Finally, we can calculate the angular momentum of the system about their center of mass:
Angular momentum = (Moment of inertia1 + Moment of inertia2) × angular velocity
Now, let's calculate the total kinetic energy of the system. The kinetic energy of each skater can be calculated using the formula:
Kinetic energy = (1/2) × mass × velocity^2
Since they are rotating, their velocity is given by:
Velocity = angular velocity × radius
The total kinetic energy is the sum of the kinetic energies of both skaters.
Now, let's plug in the values and calculate the angular momentum and total kinetic energy of the system.
To find the angular momentum of the system about their center of mass, we can use the formula:
L = I * ω
where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
The moment of inertia can be calculated using the formula:
I = m1 * r1^2 + m2 * r2^2
where m1 and m2 are the masses of the ice skaters and r1 and r2 are the distances from the center of mass to each skater.
First, let's calculate the moment of inertia. Given:
m1 = 15 kg
m2 = 50 kg
r1 = r2 = 4.6 m
I = (15 kg) * (4.6 m)^2 + (50 kg) * (4.6 m)^2
I = 1035.8 kg·m^2 + 3452 kg·m^2
I ≈ 4487.8 kg·m^2
Next, we need to find the angular velocity, ω. Since one revolution is completed in 7.6 seconds, we can calculate ω using the formula:
ω = 2π / T
where T is the time period.
ω = 2π / 7.6 s
ω ≈ 0.823 rad/s
Now we can calculate the angular momentum:
L = (4487.8 kg·m^2) * (0.823 rad/s)
L ≈ 3693.5 kg·m^2/s
Answer: The angular momentum of the system about their center of mass is approximately 3693.5 J·s.
To find the total kinetic energy of the system, we can use the formula:
K = (1/2) * (m1 * v1^2 + m2 * v2^2)
where K is the kinetic energy, m1 and m2 are the masses of the ice skaters, and v1 and v2 are the velocities of each skater. Given that the skaters are rotating in a circle, the velocity can be calculated using the formula:
v = ω * r
where ω is the angular velocity and r is the distance from the center of mass to each skater.
First, let's calculate the velocities. Given:
ω = 0.823 rad/s
r1 = r2 = 4.6 m
v1 = (0.823 rad/s) * (4.6 m)
v1 ≈ 3.785 m/s
v2 = (0.823 rad/s) * (4.6 m)
v2 ≈ 3.785 m/s
Now we can calculate the kinetic energy:
K = (1/2) * (15 kg * (3.785 m/s)^2 + 50 kg * (3.785 m/s)^2)
K = (1/2) * (15 kg * 14.31 m^2/s^2 + 50 kg * 14.31 m^2/s^2)
K = (1/2) * (214.65 kg·m^2/s^2 + 715.5 kg·m^2/s^2)
K = (1/2) * 930.15 kg·m^2/s^2
K ≈ 465.08 J
Answer: The total kinetic energy of the system is approximately 465.08 J.