A uniform solid cylinder of mass M = 6 kg is rolling without slipping along a horizontal surface. The velocity of its center of mass is 27.6 m/s. Calculate its energy
To calculate the energy of the rolling cylinder, we need to consider both its translational kinetic energy and its rotational kinetic energy.
1. Translational Kinetic Energy:
The translational kinetic energy (KE translational) is given by the formula:
KE translational = (1/2) * M * v^2
where M is the mass of the cylinder and v is the velocity of its center of mass.
Substituting the given values:
KE translational = (1/2) * 6 kg * (27.6 m/s)^2
KE translational = 2277.12 J
2. Rotational Kinetic Energy:
The rotational kinetic energy (KE rotational) is given by the formula:
KE rotational = (1/2) * I * ω^2
where I is the moment of inertia of the cylinder and ω is its angular velocity.
For a solid cylinder rolling without slipping, we have a known relationship between the translational and rotational variables:
v = ω * R
where R is the radius of the cylinder.
Rearranging the equation, we get:
ω = v / R
The moment of inertia of a solid cylinder about its central axis is given by:
I = (1/2) * M * R^2
Substituting the given values:
I = (1/2) * 6 kg * (R^2)
Note: The radius (R) is not provided in the question. You will need to have the radius of the cylinder to proceed further and calculate KE rotational.
Once you have the radius (R), substitute it in the equation and calculate KE rotational.
3. Total Energy:
The total energy (KE total) of the cylinder is the sum of its translational kinetic energy and rotational kinetic energy:
KE total = KE translational + KE rotational
Substitute the calculated values of KE translational and KE rotational, and you will have the energy of the rolling cylinder.
To calculate the energy of the rolling cylinder, we need to consider both its translational kinetic energy and its rotational kinetic energy.
1. Translational kinetic energy:
The translational kinetic energy is given by the formula:
KE_translational = (1/2) * mass * velocity^2
In this case, the mass of the cylinder is given as M = 6 kg and the velocity of its center of mass is given as 27.6 m/s.
Plugging in these values into the formula, we have:
KE_translational = (1/2) * 6 kg * (27.6 m/s)^2
Calculating this value, we get:
KE_translational = 1/2 * 6 kg * 761.76 m^2/s^2
KE_translational = 2285.28 J
2. Rotational kinetic energy:
The rotational kinetic energy is given by the formula:
KE_rotational = (1/2) * moment of inertia * angular velocity^2
For a solid cylinder rolling without slipping, the moment of inertia is given as:
I = (1/2) * mass * radius^2
In this case, the mass of the cylinder is given as M = 6 kg, and the radius of the cylinder is not provided. Therefore, we cannot calculate the exact value of the rotational kinetic energy without knowing the radius of the cylinder.
However, we can still provide the formula for rotational kinetic energy using the given values. It would be as follows:
KE_rotational = (1/2) * (1/2 * mass * radius^2) * (v/r)^2
Substituting the values we have:
KE_rotational = (1/2) * (1/2 * 6 kg * radius^2) * (27.6 m/s / radius)^2
Simplifying the equation, we get:
KE_rotational = (3/8) * 6 kg * radius^2 * (27.6 m/s / radius)^2
Therefore, without knowing the radius of the cylinder, we cannot calculate the exact value of the rotational kinetic energy.
Overall, the energy of the rolling cylinder is the sum of its translational kinetic energy and rotational kinetic energy.
Total Energy = KE_translational + KE_rotational
Please provide the value of the radius of the cylinder to compute the rotational kinetic energy accurately.
KE = KE (translational) + KE (rotational)=
=m•v²/2 + Iω²/2 =
= m•v²/2 +0.5 (m•R²/2)(v²/R²) =
=0.75 • m•v².