A spherical ball rolls without slipping determine the percentage of rotational kinetic energy
KE
Total
=
2
1
IW
2
+
2
1
MV
2
=
2
1
×
5
2
MR
2
W
2
+
2
1
MV
2
=
5
1
MR
2
W
2
+
2
1
MV
2
=
5
1
MV
2
+
2
1
MV
2
=
10
7
MV
2
KE
rot
=
2
1
IW
2
=
5
1
MV
2
KE
Total
KE
rot
=
10
7
MV
2
5
1
MV
2
=
7
2
.
To determine the percentage of rotational kinetic energy of a spherical ball that is rolling without slipping, we need to consider both the translational kinetic energy (KE_trans) and rotational kinetic energy (KE_rot).
The translational kinetic energy can be calculated using the formula:
KE_trans = (1/2) * m * v^2,
where m is the mass of the ball and v is its linear velocity.
The rotational kinetic energy can be calculated using the formula:
KE_rot = (1/2) * I * ω^2,
where I is the moment of inertia of the ball and ω is its angular velocity.
For a spherical ball rolling without slipping, the relationship between the linear velocity and angular velocity is:
v = ω * r,
where r is the radius of the ball.
We can substitute this relationship into the formula for translational kinetic energy to get:
KE_trans = (1/2) * m * (ω * r)^2 = (1/2) * m * ω^2 * r^2,
where we replace v^2 with (ω * r)^2.
Now, we need to determine the percentage of rotational kinetic energy, which is given by:
% KE_rot = (KE_rot / (KE_rot + KE_trans)) * 100.
Substituting the formulas we derived earlier, we get:
% KE_rot = ((1/2) * I * ω^2) / ((1/2) * m * ω^2 * r^2 + (1/2) * I * ω^2) * 100,
Simplifying the equation further:
% KE_rot = (I / (m * r^2 + I)) * 100.
Therefore, the percentage of rotational kinetic energy for a spherical ball rolling without slipping is given by the ratio of the moment of inertia (I) to the sum of the moment of inertia and the mass (m) multiplied by the square of the radius (r), and multiplied by 100.