The total kinetic energy of a solid sphere rolling across a horizontal surface is 15 J.
(a) What is its speed?
(b) What percent of its total kinetic energy is translational kinetic energy?
(Show your entire solution)
(a) To find the speed of the solid sphere, we first need to analyze the total kinetic energy of the sphere. The total kinetic energy of a solid sphere rolling across a horizontal surface is composed of two parts: translational kinetic energy and rotational kinetic energy. The sum of these two types of kinetic energy is equal to the total kinetic energy of the sphere.
The translational kinetic energy (K_t) is given by the formula:
K_t = (1/2) * m * v^2
where m is the mass of the sphere and v is its speed.
The rotational kinetic energy (K_r) of a solid sphere is given by the formula:
K_r = (1/2) * I * ω^2
where I is the moment of inertia of the sphere and ω is its angular velocity. For a solid sphere, the moment of inertia I is given by the formula:
I = (2/5) * m * R^2
where R is the radius of the sphere.
The angular velocity ω is related to the linear speed v by the formula:
ω = v/R
Now we can substitute the expressions for I and ω into the rotational kinetic energy formula:
K_r = (1/2) * ((2/5) * m * R^2) * (v/R)^2
K_r = (1/5) * m * v^2
Now we have expressions for the translational and rotational kinetic energy of the sphere in terms of the mass and speed:
K_t = (1/2) * m * v^2
K_r = (1/5) * m * v^2
Since the total kinetic energy K_total is the sum of these two components, we have:
K_total = K_t + K_r
15 J = (1/2) * m * v^2 + (1/5) * m * v^2
Now we can solve for the speed v. First, we factor out the m * v^2 term:
15 J = m * v^2 * (1/2 + 1/5)
15 J = m * v^2 * (7/10)
To isolate the v^2 term, we divide both sides by m * (7/10):
(15 J) / (m * (7/10)) = v^2
At this point, we cannot find the exact value of the speed v, since we don't know the mass of the sphere m. The speed of the sphere depends on its mass. However, we have found an expression for the speed in terms of the mass:
v^2 = (15 J) / (m * (7/10))
(b) To find the percentage of total kinetic energy that is translational kinetic energy, we can divide the translational kinetic energy by the total kinetic energy and multiply by 100%.
Percentage = (K_t / K_total) * 100%
From part (a), we found that the total kinetic energy is given by:
K_total = m * v^2 * (7/10)
And we know that the translational kinetic energy is given by:
K_t = (1/2) * m * v^2
Now we can substitute these expressions into the percentage formula:
Percentage = (((1/2) * m * v^2) / (m * v^2 * (7/10))) * 100%
Notice that the m * v^2 terms cancel out:
Percentage = ((1/2) / (7/10)) * 100%
Now we can simplify the expression:
Percentage = (5/7) * 100%
Percentage ≈ 0.714 * 100%
Percentage ≈ 71.4%
So about 71.4% of the total kinetic energy of the sphere is translational kinetic energy.
To find the speed of the solid sphere, we need to use the formula for the total kinetic energy of a rolling object.
(a) The total kinetic energy of a rolling object is given by the sum of its translational kinetic energy and rotational kinetic energy:
KE_total = KE_translational + KE_rotational
We know that the total kinetic energy, KE_total, is 15 J.
Substituting the values into the equation, we get:
15 = KE_translational + KE_rotational ........ (equation 1)
We also know that for a solid sphere, the rotational kinetic energy is given by:
KE_rotational = (2/5) * (1/2) * m * R^2 * ω^2
where m is the mass of the sphere, R is the radius of the sphere, and ω is the angular velocity.
Since our sphere is rolling without slipping, the angular velocity, ω, can be related to the linear velocity, v, by the equation:
v = ω * R ........ (equation 2)
Now, let's solve for the speed.
We have two unknowns: KE_translational and v. To solve for both, we need an additional equation relating them.
The translational kinetic energy is given by:
KE_translational = (1/2) * m * v^2
Substituting the above equation into equation 1, we have:
15 = (1/2) * m * v^2 + (2/5) * (1/2) * m * R^2 * ω^2 ........ (equation 3)
Substituting equation 2 into equation 3:
15 = (1/2) * m * v^2 + (2/5) * (1/2) * m * R^2 * (v/R)^2
15 = (1/2) * m * v^2 + (1/5) * m * v^2
Now, let's solve this equation for v:
15 = (1/2 + 1/5) * m * v^2
15 = (7/10) * m * v^2
Dividing both sides by (7/10) * m, we get:
v^2 = (10/7) * (15 / m)
v^2 = (150 / 7) * (1 / m)
v^2 = (150 / 7) * (1 / (4/3) * π * R^3 * ρ)
Here, ρ is the density of the sphere.
Now, we can find v by taking the square root:
v = sqrt((150 / 7) * (1 / (4/3) * π * R^3 * ρ)).
Please provide the values of R (radius), m (mass), and ρ (density) to calculate the speed.
(b) To find the percentage of the total kinetic energy that is translational kinetic energy, we can use the equation:
Percent_translational = (KE_translational / KE_total) * 100
Substituting the values, we get:
Percent_translational = (KE_translational / 15) * 100
To calculate the percentage, we need to know the value for KE_translational, which we can find using the calculated value for v.
To solve this problem, we need to apply the formulas for kinetic energy and translational kinetic energy.
(a) To find the speed of the rolling sphere, we can equate the total kinetic energy to the sum of translational and rotational kinetic energies.
The formula for the total kinetic energy of a rolling sphere is given by:
KE_total = KE_translation + KE_rotation
However, we know that the rotational kinetic energy of a solid sphere rolling on a horizontal surface is equal to 2/5 times the translational kinetic energy:
KE_rotation = (2/5) * KE_translation
We can substitute this equation into the formula for the total kinetic energy:
KE_total = KE_translation + (2/5) * KE_translation
Given that the total kinetic energy (KE_total) is 15 J, we can set up the equation as:
15 J = KE_translation + (2/5) * KE_translation
Now, let's solve for KE_translation:
15 J = KE_translation + (2/5) * KE_translation
15 J = (1 + 2/5) * KE_translation
15 J = (5/5 + 2/5) * KE_translation
15 J = (7/5) * KE_translation
Now, divide both sides of the equation by (7/5) to solve for KE_translation:
15 J / (7/5) = KE_translation
To simplify, invert (7/5) and multiply:
15 J * (5/7) = KE_translation
KE_translation = 75/7 J
Now that we have the translational kinetic energy (KE_translation), we can find the speed (v) using the formula for translational kinetic energy:
KE_translation = (1/2) * m * v^2
Where:
m = mass of the sphere
However, the mass of the sphere is not given in the problem statement. Therefore, without this information, we cannot determine the speed.
(b) To find the percentage of total kinetic energy that is translational kinetic energy, we can use the formulas we previously used.
The percentage is equal to the ratio of translational kinetic energy (KE_translation) to the total kinetic energy (KE_total), multiplied by 100:
Percentage = (KE_translation / KE_total) * 100
Substituting the values we found above:
Percentage = (KE_translation / 15 J) * 100
Percentage = ((75/7) J / 15 J) * 100
Simplifying further:
Percentage = (75/7) / 15 * 100
Percentage = (75/7) * (1/15) * 100
Percentage ≈ 33.93% (rounded to two decimal places)
Therefore, approximately 33.93% of the total kinetic energy is translational kinetic energy.