A bicycle has wheels of radius 0.32 m. Each wheel has a roatational inertia of 0.080 kg•m^2 about its axle. The total mass of the bicycle including the wheels and the rider is 79 kg. When coasting at constant speed, what fraction of the total kinetic energy of the bicycle (including rider) is the rotational kinetic energy of the wheels?
The rotational kinetic energy of one wheel is
1/2 * I * w^2
where I is the moment of inertia of the wheel, w is the angular speed of the wheel.
The total kinetic energy of the system is the sum of the rotational kinetic energies of each wheel, plus the translational kinetic energy of the bicycle:
K = 1/2*I*w^2 + 1/2*I*w^2 + 1/2 *m*v^2
v = w^2*r^2
where v is the translational speed of the bicycle, and r is the radius of the bicycle's tire
K = I*w^2 + 1/2*m*w^2*r^2
the fraction is (I*w^2) / (1/2*m*w^2*r^2)
To find the fraction of the total kinetic energy that is the rotational kinetic energy of the wheels, we need to calculate the total kinetic energy and the rotational kinetic energy separately.
The total kinetic energy (KE_total) of the bicycle and rider can be calculated using the formula:
KE_total = (1/2) * mass * velocity^2
The rotational kinetic energy (KE_rotational) of each wheel can be calculated using the formula:
KE_rotational = (1/2) * rotational inertia * rotational velocity^2
Given:
- Radius of the wheels (r) = 0.32 m
- Rotational inertia of each wheel (I) = 0.080 kg•m^2
- Mass of the bicycle and rider (m) = 79 kg
First, let's find the velocity. Since the bicycle is coasting at constant speed, the linear velocity (v) is the same as the velocity of the wheels.
The linear velocity (v) can be calculated using the formula:
v = r * angular velocity
Since the wheels are rotating about their axle, the angular velocity is constant.
Now, we can calculate angular velocity using the formula:
angular velocity = 2π / time taken for one rotation
Next, we can calculate the linear velocity (v):
v = r * angular velocity = 0.32 m * angular velocity
Now we need to calculate the total kinetic energy (KE_total):
KE_total = (1/2) * mass * velocity^2 = (1/2) * 79 kg * (0.32 m * angular velocity)^2
Finally, let's calculate the rotational kinetic energy of the wheels (KE_rotational):
KE_rotational = (1/2) * rotational inertia * rotational velocity^2 = (1/2) * 0.080 kg•m^2 * angular velocity^2
To find the fraction of the total kinetic energy that is the rotational kinetic energy of the wheels, we can divide the rotational kinetic energy of the wheels by the total kinetic energy:
Fraction = (KE_rotational / KE_total) * 100%
Substituting the values and calculating the fraction, we can find the answer.
To find the fraction of the total kinetic energy of the bicycle that is due to the rotational kinetic energy of the wheels, we first need to calculate the total kinetic energy of the bicycle.
The total kinetic energy (KE) of an object is given by the formula:
KE = (1/2)mv^2
where m is the mass of the object and v is its speed. In this case, the total mass of the bicycle, including the wheels and the rider, is 79 kg.
Next, we need to calculate the rotational kinetic energy (Kerot) of the wheels. The formula for the rotational kinetic energy is:
Kerot = (1/2)Iω^2
where I is the rotational inertia of the wheels and ω is the angular velocity of the wheels. The rotational inertia of each wheel is given as 0.080 kg•m^2.
Since the bicycle is coasting at constant speed, the linear speed of the bicycle is the same as the speed of the rider. However, since the wheels are rotating, their angular velocity is related to the linear speed by the equation:
v = ωr
where r is the radius of the wheel. In this case, the radius of the wheel is given as 0.32 m.
To find the angular velocity of the wheels (ω), we rearrange the equation:
ω = v / r
Now we can substitute the given values into the formulas to calculate the total kinetic energy of the bicycle (KE) and the rotational kinetic energy of the wheels (Kerot).
KE = (1/2)mv^2
Kerot = (1/2)Iω^2
Finally, we can find the fraction of the total kinetic energy that is due to the rotational kinetic energy of the wheels by dividing the rotational kinetic energy by the total kinetic energy:
Fraction = Kerot / KE
By following these steps and substituting the given values, we can find the desired fraction.