A pitcher can throw a ball towards the home plate at 120km per hour. When thrown, the ball curves. The curveball has a spin of 1250rpm. The ball has a mass of 100g and a radius of 2.5cm

Calculate the percentage of total KE that is in rotational KE

translationKE=1/2 m v^2 change m to kg, v to m/s

rotational KE=(w^2*r= 2/5 M r^2*(2pi*1250/60)^2

total: add them

decimal percent: rotatioal/total

To calculate the percentage of total kinetic energy (KE) that is in rotational KE, we need to know the total kinetic energy of the ball and the rotational kinetic energy of the ball.

First, let's calculate the total kinetic energy using the formula for kinetic energy:

KE = 0.5 * mass * velocity^2

Given:
Mass (m) = 100g = 0.1kg
Velocity (v) = 120 km/h = 120000 m/3600 s = 33.33 m/s

Plugging in these values:

KE = 0.5 * 0.1kg * (33.33 m/s)^2
= 0.5 * 0.1kg * 1110.89 m^2/s^2
= 55.54 J (Joules)

Now, let's calculate the rotational kinetic energy using the formula:

Rotational KE = 0.5 * moment of inertia * angular velocity^2

The moment of inertia (I) of a solid sphere is given by:

I = (2/5) * mass * radius^2

Given:
Mass (m) = 100g = 0.1kg
Radius (r) = 2.5cm = 0.025m
Angular velocity (ω) = 2π * rpm/60

Plugging in these values:

I = (2/5) * 0.1kg * (0.025m)^2
= 0.00008333 kg.m^2 (rounded to 5 decimal places)

ω = 2π * 1250/60
= 130.90 rad/s (rounded to 2 decimal places)

Rotational KE = 0.5 * 0.00008333 kg.m^2 * (130.90 rad/s)^2
= 0.5 * 0.00008333 kg.m^2 * 17124.81 rad^2/s^2
= 710.18 J (Joules) (rounded to 2 decimal places)

To find the percentage of total KE that is in rotational KE, divide the rotational KE by the total KE and multiply by 100:

Percentage = (Rotational KE / Total KE) * 100
= (710.18 J / 55.54 J) * 100
= 1278.55%

Therefore, the percentage of total kinetic energy that is in rotational kinetic energy is approximately 1278.55%.