A bowling ball has a mass of 7.0 kg, a moment of inertia of 2.8 × 10−2 kg⋅m2 and a radius of 0.10 m. If it rolls down the lane without slipping at a linear speed of 4.0 m/s, what is its total kinetic energy?

Ke = (1/2) m v^2 + (1/2)I omega^2

you have everything there but omega, the angular speed, but you know that
omega = v/R = 4.0/0.10 = 40 rad/second

To find the total kinetic energy of the bowling ball, we need to consider both its translational kinetic energy and rotational kinetic energy.

1. Translational kinetic energy can be calculated using the formula:
KE_translational = (1/2) * m * v^2
where m is the mass of the bowling ball and v is its linear speed.

Substituting the given values:
KE_translational = (1/2) * 7.0 kg * (4.0 m/s)^2
= (1/2) * 7.0 kg * 16.0 m^2/s^2
= 56.0 kg⋅m^2/s^2

2. Rotational kinetic energy can be calculated using the formula:
KE_rotational = (1/2) * I * ω^2
where I is the moment of inertia of the bowling ball and ω is its angular speed.

Since the ball is rolling without slipping, the linear speed is related to the angular speed as:
v = ω * r
where r is the radius of the bowling ball.

Rearranging the equation, we can solve for ω:
ω = v / r
ω = 4.0 m/s / 0.10 m
ω = 40 rad/s

Substituting the values:
KE_rotational = (1/2) * (2.8 × 10^-2 kg⋅m^2) * (40 rad/s)^2
= (1/2) * (2.8 × 10^-2 kg⋅m^2) * 1600 rad^2/s^2
= 22.4 kg⋅m^2/s^2

3. The total kinetic energy is the sum of the translational and rotational kinetic energies:
Total KE = KE_translational + KE_rotational
= 56.0 kg⋅m^2/s^2 + 22.4 kg⋅m^2/s^2
= 78.4 kg⋅m^2/s^2

Therefore, the total kinetic energy of the bowling ball is 78.4 kg⋅m^2/s^2.

To find the total kinetic energy of the bowling ball, you need to consider two components: translational kinetic energy and rotational kinetic energy.

1. Translational Kinetic Energy:
The translational kinetic energy is given by the formula: KE_translational = (1/2) * mass * velocity^2

Given:
Mass of the bowling ball (m) = 7.0 kg
Linear speed (v) = 4.0 m/s

Using the formula, we can calculate the translational kinetic energy:
KE_translational = (1/2) * 7.0 kg * (4.0 m/s)^2 = 0.5 * 7.0 kg * 16.0 m^2/s^2 = 56.0 J

2. Rotational Kinetic Energy:
The rotational kinetic energy is given by the formula: KE_rotational = (1/2) * moment of inertia * angular velocity^2

Given:
Moment of inertia (I) = 2.8 × 10^(-2) kg⋅m^2

To calculate the angular velocity (ω), we need the formula:
ω = v / r

where:
v = linear speed = 4.0 m/s
r = radius = 0.10 m

Calculating the angular velocity:
ω = 4.0 m/s / 0.10 m = 40 rad/s

Now we can calculate the rotational kinetic energy:
KE_rotational = (1/2) * (2.8 × 10^(-2) kg⋅m^2) * (40 rad/s)^2 = 22.4 J

Finally, to find the total kinetic energy of the bowling ball, we add the translational and rotational kinetic energies together:
Total KE = KE_translational + KE_rotational
Total KE = 56.0 J + 22.4 J = 78.4 J

Therefore, the total kinetic energy of the bowling ball is 78.4 Joules.

You might start here:

http://www.dummies.com/education/science/physics/how-to-calculate-rotational-kinetic-energy/