(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.
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)
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