A spoked wheel with a radius of 44.0 cm and a mass of 1.90 kg is mounted horizontally on frictionless bearings. JiaJun puts his 0.550-kg guinea pig on the outer edge of the wheel. The guinea pig begins to run along the edge of the wheel with a speed of 18.0 cm/s with respect to the ground. What is the angular velocity of the wheel? Assume the spokes of the wheel have negligible mass.
rad/s
Honestly, I don't know how to calculate this, I have been trying to figure out but I have been guessing, and am sure they aren't write because they gave me wrong answers. Need help thank you in advance.
Since the bearings are frictionless, the total angular momentum of wheel and guinea pig remains zero, as they rotate in opposite directions.
0.18 m/s*0.550kg*0.44m
= 0.04356 kg m^2/s
= -Iwheel*w
The moment of inertia of the wheel is
Iwheel = M*R^2 = 1.90*(0.44)^2
= 0.3678 kg*m^2. Therefore
w = -0.118 rad/s
The minus sign indicates that it is turning in the opposite direction from the guinea pig.
Thank you so much; I really appreciate it!
To calculate the angular velocity of the wheel, we'll use the principle of conservation of angular momentum. Angular momentum is defined as the product of moment of inertia and angular velocity.
First, let's find the moment of inertia of the wheel. The moment of inertia of a solid disc rotating about its center is given by the formula:
I = (1/2) * m * r^2
where
I = moment of inertia
m = mass of the wheel
r = radius of the wheel
Substituting the given values:
I = (1/2) * 1.90 kg * (0.44m)^2
I ≈ 0.178 kg·m^2
Now, since the guinea pig is running along the outer edge of the wheel, its perpendicular distance from the center of rotation is equal to the radius of the wheel, i.e., 0.44 m.
According to the conservation of angular momentum, the initial angular momentum of the system (wheel + guinea pig) is equal to the final angular momentum. The initial angular momentum (Li) is given by:
Li = I * ωi
where
ωi = initial angular velocity
The final angular momentum (Lf) is given by:
Lf = I * ωf
where
ωf = final angular velocity (which we need to find)
Since angular momentum is conserved, Li = Lf. Therefore:
I * ωi = I * ωf
Canceling out the moment of inertia on both sides:
ωi = ωf
So the initial angular velocity (ωi) of the system is equal to the final angular velocity (ωf) of the wheel.
Given that the guinea pig is running along the edge of the wheel with a speed of 18.0 cm/s with respect to the ground, we can convert it to meters per second (m/s) by dividing by 100:
v = 18.0 cm/s / 100
v = 0.18 m/s
The linear velocity (v) at the outer edge of the wheel is related to the angular velocity (ω) by the formula:
v = ω * r
where
r = radius of the wheel
Rearranging the formula:
ω = v / r
Substituting the values:
ω = 0.18 m/s / 0.44 m
ω ≈ 0.409 rad/s
So, the angular velocity of the wheel is approximately 0.409 rad/s.