Hello,
The question is: A solid ball rolls along the floor with a constant linear speed v. Find the fraction of its total kinetic energy that is in the form of rotational kinetic energy about the center of the ball.
Krot/Ktotal = ?
The answer I keep getting is 3/1, but the online homework system says "Your response differs from the correct answer by orders of magnitude.". I know the I for a solid sphere is I = 2/3 mr^2, but I'm still stuck.
Thank you for the help.
KE=1/2mv^2 + 1/2 *2/3 mr^2*w^2
but v=rw
so w^2=v^2/r^2
KE=1/2mv^2+1/3 mv^2 = 5/6mv^2
what fraction is rotational?
1/3/(5/6)=2/5 or .4 is rotational
check my math.
KE=1/2mv^2 + 1/2 *2/5 mr^2*w^2 ..not 2/3 mr^2*w^2 its solid sphere not a hollow sphere.
so the fraction would be 2/7 not 2/5 as the final answer. But bob's approach seems correct.
To find the fraction of the total kinetic energy that is in the form of rotational kinetic energy, you can use the equation for the kinetic energy of rotation of a solid sphere. The equation is given as:
Krot = (1/2)Iω^2
Where Krot is the rotational kinetic energy, I is the moment of inertia, and ω is the angular velocity.
The moment of inertia for a solid sphere is given as I = (2/5)mr^2, not I = (2/3)mr^2.
First, let's find the linear speed in terms of the angular velocity. The linear speed of the ball is v, and it is related to the angular speed by v = rω, where r is the radius of the sphere. Rearranging the equation, we have ω = v/r.
Substituting this value of ω into the equation for rotational kinetic energy, we get:
Krot = (1/2)(2/5)mr^2(v/r)^2
= (1/5)mv^2
Now, let's find the total kinetic energy of the rolling ball. The total kinetic energy consists of the kinetic energy due to linear motion and the rotational kinetic energy. Since the ball has a constant linear speed, its kinetic energy of linear motion is given by Klin = (1/2)mv^2.
The total kinetic energy Ktotal is then:
Ktotal = Klin + Krot
= (1/2)mv^2 + (1/5)mv^2
= (3/10)mv^2
To find the fraction of the total kinetic energy that is in the form of rotational kinetic energy, we divide the rotational kinetic energy by the total kinetic energy:
Krot/Ktotal = ((1/5)mv^2) / ((3/10)mv^2)
= (1/5)/(3/10)
= 2/3
So, the correct answer is Krot/Ktotal = 2/3, not 3/1. It seems like you made a mistake when calculating the rotational kinetic energy fraction.
To find the fraction of the total kinetic energy that is in the form of rotational kinetic energy, we need to consider the rotational kinetic energy and the total kinetic energy of the solid ball.
Let's start by considering the rotational kinetic energy (Krot) of the solid ball. The rotational kinetic energy is given by the formula:
Krot = (1/2) I ω^2
Where:
Krot is the rotational kinetic energy
I is the moment of inertia of the ball
ω is the angular velocity of the ball
In this case, the ball is rolling without slipping, which means the linear speed (v) at the surface of the ball is related to the angular velocity by the equation:
v = R ω
Where:
v is the linear speed of the ball
R is the radius of the ball
ω is the angular velocity of the ball
Now, let's substitute the value of ω from the above equation into the formula for rotational kinetic energy:
Krot = (1/2) I (v/R)^2
The total kinetic energy (Ktotal) of the solid ball is given by the formula:
Ktotal = (1/2) m v^2
Where:
Ktotal is the total kinetic energy
m is the mass of the ball
v is the linear speed of the ball
Now, let's find the fraction of rotational kinetic energy to total kinetic energy by dividing Krot by Ktotal:
Krot/Ktotal = [(1/2) I (v/R)^2] / [(1/2) m v^2]
= (I/R^2) / (m v^2)
Substituting the value of I for a solid sphere (I = (2/3) m R^2):
Krot/Ktotal = [(2/3) m R^2 / R^2] / (m v^2)
= (2/3) / v^2
Therefore, the correct answer for the fraction of the total kinetic energy that is in the form of rotational kinetic energy is (2/3) / v^2.
Now, let's check your answer of 3/1. The fraction 3/1 doesn't make sense because it implies that the rotational kinetic energy is three times greater than the total kinetic energy, which is not possible. It seems like there might be a calculation error in your solution.
Please double-check your calculations and verify if you made any mistakes.