Calculate the angular momentum and kinetic energy of a solid uniform sphere with a radius of 0.120 m and a mass of 14.0 kg if it is rotating at 6.00 rad/s about an axis through its center..
Well, I would start calculating those values for you, but I'm afraid I would just end up going in circles. Get it? Because it's rotating? Alright, alright, I'll give it a shot.
The angular momentum (L) of a rotating object is given by the formula L = Iω, where I is the moment of inertia and ω is the angular velocity.
The moment of inertia (I) for a solid uniform sphere can be calculated as I = (2/5) * m * r^2, where m is the mass of the sphere and r is the radius.
Substituting the given values into the formula, we get:
I = (2/5) * 14.0 kg * (0.120 m)^2
Evaluating this expression, we find that I = 0.24192 kg·m².
Now we can calculate the angular momentum:
L = I * ω = 0.24192 kg·m² * 6.00 rad/s
Evaluating this expression, we find that L = 1.45152 kg·m²/s. That's a lot of spinning!
As for the kinetic energy (K) of the rotating sphere, it can be calculated as K = (1/2) * I * ω^2.
Substituting the given values into the formula, we get:
K = (1/2) * 0.24192 kg·m² * (6.00 rad/s)^2
Evaluating this expression, we find that K = 4.36368 J. That's quite an energetic little sphere!
So, to recap:
The angular momentum of the sphere is 1.45152 kg·m²/s, and its kinetic energy is 4.36368 J. Now, if only we could harness all that spinning power for something useful... like making a really cool carnival ride!
To calculate the angular momentum and kinetic energy of a rotating solid uniform sphere, we need to use the formulas related to rotational motion.
1. Angular momentum (L) is given by the equation:
L = I * ω
where I represents the moment of inertia and ω represents the angular velocity. The moment of inertia for a solid uniform sphere rotating about an axis through its center is given by the formula:
I = (2/5) * m * r^2
where m represents the mass of the sphere and r represents the radius.
2. Kinetic energy (K) is given by the equation:
K = (1/2) * I * ω^2
Now we can substitute the given values into the equations and calculate the angular momentum and kinetic energy.
Given:
Mass of the sphere (m) = 14.0 kg
Radius of the sphere (r) = 0.120 m
Angular velocity (ω) = 6.00 rad/s
1. Calculating the moment of inertia (I):
I = (2/5) * m * r^2
I = (2/5) * 14.0 kg * (0.120 m)^2
Calculate the value of I.
2. Calculating the angular momentum (L):
L = I * ω
Substitute the calculated value of I and the given value of ω into the equation.
3. Calculating the kinetic energy (K):
K = (1/2) * I * ω^2
Substitute the calculated value of I and the given value of ω into the equation.
Perform these calculations to find the values of angular momentum (L) and kinetic energy (K) for the given solid uniform sphere.
r=0.12 m
m=14 kg
Moment of inertia of sphere, I
= (2/5)mr2
=0.08064 kg-m2
Angular velocity, ω
= 6 rad/sec.
Angular momentum, L
= Iω
= 0.48384 kg-m2/s.
Rotational kinetic energy, Kr
= (1/2)Iω2
= (1/2)0.08064 62 kg-m2/s/s
= 1.45152 kg-m2/s2
To calculate the angular momentum of a rotating sphere, you can use the formula:
Angular Momentum (L) = Moment of Inertia (I) * Angular Velocity (ω)
The moment of inertia of a solid uniform sphere is given by the equation:
Moment of Inertia (I) = (2/5) * mass * radius^2
Given:
Radius (r) = 0.120 m
Mass (m) = 14.0 kg
Angular Velocity (ω) = 6.00 rad/s
Let's calculate the moment of inertia first:
I = (2/5) * m * r^2
I = (2/5) * 14.0 kg * (0.120 m)^2
I = 0.2688 kg·m²
Now, we can calculate the angular momentum:
L = I * ω
L = 0.2688 kg·m² * 6.00 rad/s
L ≈ 1.613 kg·m²/s
The angular momentum of the rotating sphere is approximately 1.613 kg·m²/s.
To calculate the kinetic energy of the rotating sphere, you can use the formula:
Kinetic Energy (KE) = (1/2) * I * ω^2
Substituting the values we already calculated:
KE = (1/2) * 0.2688 kg·m² * (6.00 rad/s)^2
KE ≈ 4.838 J
The kinetic energy of the rotating sphere is approximately 4.838 Joules.