A grinding wheel in the shape of a solid disk is 0.200 m in diameter and has a mass of 3.00 kg. The wheel is rotating at 2200 rpm about an axis through its center. (a) what is its kinetic energy? (b) How far would it have to drop in free fall to acquire the same amount of kinetic energy?
KE=1/2 I w^2
Look up I for a solid disk.
w=2PI*2200/60 rad/sec
To calculate the kinetic energy of the grinding wheel, we can use the formula:
Kinetic Energy (KE) = 0.5 * moment of inertia * angular velocity^2
(a) To find the moment of inertia (I) of the grinding wheel, we can assume it to be a solid disk. The moment of inertia of a solid disk is given by:
I = (1/2) * mass * radius^2
Given:
Mass (m) = 3.00 kg
Diameter (d) = 0.200 m
Convert the diameter to radius:
Radius (r) = d/2 = 0.200/2 = 0.100 m
Now, we can calculate the moment of inertia:
I = (1/2) * m * r^2
I = (1/2) * 3.00 kg * (0.100 m)^2
I = 0.015 kg·m^2
Next, we have the angular velocity (ω) in units of revolutions per minute (rpm). We need to convert it to radians per second (rad/s). There are 2π radians in one revolution, and 60 seconds in one minute:
ω = (2π * rpm) / 60
ω = (2π * 2200 rpm) / 60
ω = 230.52 rad/s (rounded to two decimal places)
Now, we can substitute the values into the kinetic energy formula:
KE = 0.5 * I * ω^2
KE = 0.5 * 0.015 kg·m^2 * (230.52 rad/s)^2
KE = 194.6 Joules (rounded to one decimal place)
Thus, the kinetic energy of the grinding wheel is 194.6 Joules.
(b) To find how far the grinding wheel would have to drop in free fall to acquire the same amount of kinetic energy, we can use the concept of potential energy conservation. The potential energy (PE) gained by an object when it falls is equal to the kinetic energy it acquires.
PE = m * g * h
Rearranging the equation, we can solve for the height (h):
h = KE / (m * g)
Given:
Mass (m) = 3.00 kg
Acceleration due to gravity (g) = 9.8 m/s^2
Kinetic Energy (KE) = 194.6 Joules
Substituting the values into the equation:
h = 194.6 J / (3.00 kg * 9.8 m/s^2)
h ≈ 6.68 m (rounded to two decimal places)
Thus, the grinding wheel would have to drop approximately 6.68 meters in free fall to acquire the same amount of kinetic energy.
To find the answers to these questions, we need to use the formulas for kinetic energy and potential energy.
(a) To find the kinetic energy (KE) of the grinding wheel, we can use the formula:
KE = (1/2) * I * ω²
where I is the moment of inertia and ω is the angular velocity.
1. First, let's find the moment of inertia (I) of the wheel. Since it is a solid disk, the moment of inertia is given by:
I = (1/2) * m * r²
where m is the mass of the wheel and r is the radius.
Given:
m = 3.00 kg
r = 0.200 m (since the diameter is 0.200 m, the radius is half of that)
Substituting the values, we get:
I = (1/2) * 3.00 kg * (0.200 m)²
I = 0.060 kg * m²
2. Next, we need to convert the angular velocity from rpm to rad/s. Since 1 rotation equals 2π radians, we multiply the angular velocity (in rpm) by (2π/60) to get it in rad/s.
Angular velocity (ω) in rad/s = (2200 rpm) * (2π/60)
angular velocity (ω) = 230.062 rad/s (approximately)
Now, we can calculate the kinetic energy (KE) of the grinding wheel using the formula:
KE = (1/2) * I * ω²
KE = (1/2) * (0.060 kg * m²) * (230.062 rad/s)²
After performing the calculation, you'll find the value of KE in joules.
(b) To determine how far the wheel would have to drop in free fall to acquire the same amount of kinetic energy, we can use the principle of conservation of energy. The potential energy (PE) gained by an object in free fall is equal to its change in potential energy, which is given by:
ΔPE = m * g * h
where m is the mass of the wheel, g is the acceleration due to gravity, and h is the height.
We want to find the height (h) at which the potential energy gained would be equal to the kinetic energy (KE) of the wheel.
Using the equation for potential energy, we can rearrange it to solve for h:
h = KE / (m * g)
Given values:
KE (from part a) = value in joules
m = 3.00 kg
g = 9.8 m/s² (acceleration due to gravity)
Substitute the values to calculate the value of h in meters.
Please note that in part (b), the height h represents the distance the wheel would have to drop in free fall to acquire the same kinetic energy.