A solid sphere of uniform density starts from rest and rolls without slipping a distance of d = 4.4 m down a q = 20° incline. The sphere has a mass M = 4.6 kg and a radius R = 0.28 m. Of the total kinetic energy of the sphere, what fraction is translational?
KEtran/KEtotal = ?
Oh, rolling spheres, they bring back memories of my days as a circus performer! Let's calculate that fraction, shall we?
To start, we need to find the total kinetic energy of the sphere. The formula for that is KEtotal = 1/2 * I * ω^2, where I is the moment of inertia and ω is the angular velocity.
For a solid sphere rolling without slipping down an incline, the moment of inertia is given by I = 2/5 * m * R^2, where m is the mass of the sphere and R is its radius.
First, let's find the angular velocity ω. To do that, we can use the relationship between linear and angular velocity for a rolling object, which is v = ω * R, where v is the linear velocity.
In this case, the linear velocity v is given by v = √(2 * g * d * sin(q)), where g is the acceleration due to gravity, d is the distance rolled, and q is the angle of the incline.
So, we have v = ω * R, which gives us ω = v / R.
Plugging in the values, we get ω = (√(2 * g * d * sin(q))) / R.
Now, let's substitute the values we know into the formula for the moment of inertia: I = 2/5 * m * R^2.
Substituting that value for I and the value for ω into the formula for KEtotal, we get KEtotal = 1/2 * (2/5 * m * R^2) * ((√(2 * g * d * sin(q))) / R)^2.
Simplifying that expression, we get KEtotal = (1/5) * m * (g * d * sin(q)).
Now that we have the total kinetic energy, we can find the fraction that is translational. The translational kinetic energy (KEtran) is given by KEtran = 1/2 * m * v^2, where v is the linear velocity.
Substituting the value for v, we get KEtran = 1/2 * m * (√(2 * g * d * sin(q)))^2.
Simplifying that expression, we get KEtran = m * g * d * sin(q).
Now, we can find the fraction of translational kinetic energy by dividing KEtran by KEtotal:
KEtran/KEtotal = (m * g * d * sin(q)) / ((1/5) * m * (g * d * sin(q))).
Canceling out some terms, we get KEtran/KEtotal = 5/5.
Ah, the answer! It seems that the fraction of the total kinetic energy that is translational is 1. That's right, the translational energy is the whole package! I guess this sphere doesn't have any fancy spinning tricks up its sleeve.
To find the fraction of translational kinetic energy, we need to calculate both the translational and rotational kinetic energies of the sphere and then determine the ratio.
1. Translational Kinetic Energy (KEtran):
The translational kinetic energy is given by the formula:
KEtran = (1/2) * M * V^2
where M is the mass of the sphere and V is the velocity of the center of mass.
2. Rotational Kinetic Energy (KErot):
The rotational kinetic energy of a rolling sphere is given by the formula:
KErot = (1/2) * I * ω^2
where I is the moment of inertia of the sphere and ω is the angular velocity.
To calculate the translational and rotational velocities, we'll use the conservation of energy principle, which states that the initial potential energy (mgh) is equal to the final total kinetic energy (KEtotal).
3. Total Kinetic Energy (KEtotal):
The total kinetic energy consists of both translational and rotational kinetic energies:
KEtotal = KEtran + KErot
Now let's calculate the individual components step-by-step:
Step 1: Find the translational velocity (V).
To find V, we can use the conservation of energy equation:
mgh = (1/2) * M * V^2 + (1/2) * I * ω^2
Step 2: Find the rotational velocity (ω).
The rolling motion of the sphere is related to its translational velocity by the equation:
V = ω * R
Substitute this equation into the conservation of energy equation and solve for V:
mgh = (1/2) * M * V^2 + (1/2) * I * (V/R)^2
Step 3: Calculate the moment of inertia (I).
For a solid sphere, the moment of inertia is given by:
I = (2/5) * M * R^2
Substitute the given values into the equation to find I.
Step 4: Solve for V.
mgh = (1/2) * M * V^2 + (1/2) * (2/5) * M * R^2 * (V/R)^2
Now we substitute the given values:
m = 4.6 kg, g = 9.8 m/s^2, h = 4.4 m, R = 0.28 m.
Solve the equation numerically to find V.
Step 5: Calculate the translational kinetic energy (KEtran).
Once we have V, we can use the formula to calculate the translational kinetic energy again:
KEtran = (1/2) * M * V^2
Step 6: Calculate the rotational kinetic energy (KErot).
Lastly, we can calculate the rotational kinetic energy using the formula:
KErot = (1/2) * I * ω^2
Now, to find the fraction of translational kinetic energy (KEtran/KEtotal), we can divide the translational kinetic energy by the total kinetic energy and multiply by 100 to convert it to a percentage:
KEtran/KEtotal = (KEtran / (KEtran + KErot)) * 100
Plug in the values calculated in the previous steps and you will get the fraction of translational kinetic energy.
To find the fraction of the total kinetic energy that is translational, we need to calculate both the translational kinetic energy and the total kinetic energy of the rolling sphere.
The translational kinetic energy (KEtran) is given by the formula:
KEtran = 0.5 * m * v^2
where m is the mass of the sphere and v is the velocity of the center of mass.
The total kinetic energy (KEtotal) is the sum of the translational kinetic energy and the rotational kinetic energy. For a rolling sphere, the rotational kinetic energy (KErot) can be calculated using the formula:
KErot = 0.5 * I * ω^2
where I is the moment of inertia of the sphere and ω is the angular velocity.
In this case, since the sphere is rolling without slipping, we can relate the linear velocity v and the angular velocity ω using the equation:
v = ω * R
where R is the radius of the sphere.
To determine the fraction of translational kinetic energy, we need to find both KEtran and KEtotal. Here's how you can calculate it step by step:
Step 1: Calculate the angular velocity (ω):
Using the relationship v = ω * R, we can solve for ω by rearranging the equation:
ω = v / R
Step 2: Calculate the translational kinetic energy (KEtran):
Substitute the given values into the formula for KEtran:
KEtran = 0.5 * m * v^2
Step 3: Calculate the rotational kinetic energy (KErot):
Substitute the calculated value of ω into the formula for KErot:
KErot = 0.5 * I * ω^2
Step 4: Calculate the moment of inertia (I) for a solid sphere:
The moment of inertia for a solid sphere can be calculated using the formula:
I = (2/5) * m * R^2
Step 5: Calculate the total kinetic energy (KEtotal):
The total kinetic energy is the sum of the translational and rotational kinetic energies:
KEtotal = KEtran + KErot
Step 6: Finally, calculate the fraction of translational kinetic energy:
KEtran/KEtotal = KEtran / (KEtran + KErot)
By following these steps and substituting the given values for mass (m), radius (R), distance (d), and angle (q), you can find the fraction of the total kinetic energy that is translational.