Find the percentages of the total kinetic energy associated with rotation and translation for the following objects that are rolling without slipping.
a. a uniform sphere
K rot =
b. K trans =
c. a uniform cylinder
K rot =
d. K trans =
e. a hoop
K rot =
K trans =
a. For a uniform sphere rolling without slipping, the rotational kinetic energy (Krot) is equal to 2/5 of the total kinetic energy (Ktotal). So, Krot = 2/5 * Ktotal.
b. The translational kinetic energy (Ktrans) for a uniform sphere rolling without slipping is equal to 3/5 of the total kinetic energy (Ktotal). So, Ktrans = 3/5 * Ktotal.
c. For a uniform cylinder rolling without slipping, the rotational kinetic energy (Krot) is equal to 1/2 of the total kinetic energy (Ktotal). So, Krot = 1/2 * Ktotal.
d. The translational kinetic energy (Ktrans) for a uniform cylinder rolling without slipping is equal to 1/2 of the total kinetic energy (Ktotal). So, Ktrans = 1/2 * Ktotal.
e. For a hoop rolling without slipping, all of the kinetic energy is associated with rotation. So, Krot = Ktotal and Ktrans = 0%. (But don't worry, hooping always gets people moving!)
To find the percentages of the total kinetic energy associated with rotation and translation for objects rolling without slipping, we need to use the equations for kinetic energy of rotation (Krot) and kinetic energy of translation (Ktrans).
The equations are as follows:
Krot = (1/2)Iω^2
Ktrans = (1/2)mv^2
where:
I = moment of inertia
ω = angular velocity
m = mass
v = linear velocity
Now let's calculate the percentages for each case:
a. For a uniform sphere:
The moment of inertia for a sphere rolling without slipping is given by I = (2/5)mr^2, where r is the radius.
So, Krot = (1/2)(2/5)mr^2ω^2 = (1/5)m(r^2)ω^2
Ktrans = (1/2)mv^2
The total kinetic energy is the sum of Krot and Ktrans:
Ktotal = Krot + Ktrans
To find the percentage of Krot with respect to Ktotal, we can use the formula:
Percentage of Krot = (Krot / Ktotal) * 100
b. Similarly, for Ktrans, the percentage can be found using the formula:
Percentage of Ktrans = (Ktrans / Ktotal) * 100
Let's calculate the percentages for each case:
a. For a uniform sphere:
Krot = (1/5)m(r^2)ω^2
Ktrans = (1/2)mv^2
b. For a uniform cylinder:
Krot = (1/2)Iω^2 = (1/2)(1/2)m(r^2)(v/r)^2 = (1/4)mv^2
Ktrans = (1/2)mv^2
c. For a hoop:
Krot = (1/2)Iω^2 = (1/2)(mr^2)(v/r)^2 = (1/2)mv^2
Ktrans = (1/2)mv^2
Now, let's find the percentages:
a. For a uniform sphere:
Percentage of Krot = (Krot / Ktotal) * 100 = [(1/5)m(r^2)ω^2 / (1/5)m(r^2)ω^2 + (1/2)mv^2] * 100
b. For a uniform cylinder:
Percentage of Krot = (Krot / Ktotal) * 100 = [(1/4)mv^2 / (1/4)mv^2 + (1/2)mv^2] * 100
c. For a hoop:
Percentage of Krot = (Krot / Ktotal) * 100 = [(1/2)mv^2 / (1/2)mv^2 + (1/2)mv^2] * 100
Note: The total kinetic energy is the sum of Krot and Ktrans, so Ktotal = Krot + Ktrans.
We can simplify the percentages further if we have specific values for the angular velocity (ω), linear velocity (v), and radius (r).
To find the percentages of the total kinetic energy associated with rotation and translation for objects rolling without slipping, we need to use the formulas for rotational and translational kinetic energy.
The rotational kinetic energy, K_rot, is given by:
K_rot = (1/2) * I * ω^2,
where I is the moment of inertia of the object and ω is the angular velocity.
The translational kinetic energy, K_trans, is given by:
K_trans = (1/2) * m * v^2,
where m is the mass of the object and v is its linear velocity.
Now let's calculate the percentages for each object:
a. Uniform sphere:
For a uniform sphere rolling without slipping, the moment of inertia can be calculated as I = (2/5) * m * r^2, where r is the radius of the sphere.
K_rot = (1/2) * [(2/5) * m * r^2] * ω^2
= (1/5) * m * r^2 * ω^2
K_trans = (1/2) * m * v^2
To find the percentages, we need to divide K_rot and K_trans by the total kinetic energy, which is the sum of rotational and translational energy:
K_total = K_rot + K_trans.
So the percentage of rotational kinetic energy is:
K_rot_percentage = (K_rot / K_total) * 100.
Similarly, the percentage of translational kinetic energy is:
K_trans_percentage = (K_trans / K_total) * 100.
b. Uniform cylinder:
For a uniform cylinder rolling without slipping, the moment of inertia can be calculated as I = (1/2) * m * r^2, where r is the radius of the cylinder.
K_rot = (1/2) * [(1/2) * m * r^2] * ω^2
= (1/4) * m * r^2 * ω^2
K_trans = (1/2) * m * v^2
c. Hoop:
For a hoop rolling without slipping, the moment of inertia can be calculated as I = m * r^2, where r is the radius of the hoop.
K_rot = (1/2) * (m * r^2) * ω^2
= (1/2) * m * r^2 * ω^2
K_trans = (1/2) * m * v^2
By following these calculations, you can find the percentages of the total kinetic energy associated with rotation and translation for each object.