A small motor is mounted on the axis of a space probe with its rotor (the rotating part of the motor) parallel to the axis of the probe. Its function is to control the rotational position of the probe about the axis. The moment of inertia of the probe is 6000 times that of the rotor. Initially, the probe and rotor are at rest. The motor is turned on and after some period of time, the probe is seen to have rotated by positive 32.6 degrees.
Through how many revolutions has the rotor turned?
To determine how many revolutions the rotor has turned, we need to calculate the angle through which the rotor has rotated in radians.
Given:
Angle of rotation of the probe: 32.6 degrees
We can convert this angle to radians:
1 revolution = 360 degrees = 2π radians
So, 32.6 degrees = (32.6/360) * 2π radians ≈ 0.568 radians
Now, let's find the ratio of the moment of inertia of the probe to the rotor:
Moment of inertia of the rotor = I_r
Moment of inertia of the probe = I_p = 6000 * I_r
Therefore, we can express the conservation of angular momentum as:
I_r * ω_r = I_p * ω_p
where,
ω_r = angular velocity of the rotor
ω_p = angular velocity of the probe
Since the initial angular velocities are zero, we have:
0 = I_r * ω_r - I_p * ω_p
Now, we can substitute the values:
0 = I_r * ω_r - (6000 * I_r) * ω_p
Simplifying the equation:
0 = I_r * (ω_r - 6000 * ω_p)
Given that the rotor and probe are connected, they have the same angular velocity, so we have:
0 = I_r * (1 - 6000 * ω_p)
Next, we relate the angle of rotation of the rotor to the angular velocity:
θ_r = ω_r * t
where,
θ_r = angle of rotation of the rotor
t = time
Solving for ω_r, we get:
ω_r = θ_r / t
Substituting this into the equation above, we have:
0 = I_r * (1 - (6000 * θ_p) / (I_r * t))
Now, we can rearrange the equation to solve for the angle of rotation of the rotor:
1 = 6000 * θ_p / (I_r * t)
Finally, let's solve for the angle of rotation of the rotor (in revolutions):
θ_r = ω_r * t
= (θ_r / t) * t
= θ_r
Therefore, the rotor has turned by an angle of 0.568 radians or approximately 0.091 revolutions.
To determine how many revolutions the rotor has turned, we first need to convert the angle of rotation from degrees to radians.
1 revolution is equal to 360 degrees or 2π radians.
Given that the probe has rotated by 32.6 degrees, we can convert this to radians by multiplying by the conversion factor of π/180:
32.6 degrees * (π/180) = 0.568 radians
Now, we know that the moment of inertia of the probe (I) is 6000 times that of the rotor. Let's denote the moment of inertia of the rotor as I_r.
So, I = 6000 * I_r
Now, let's apply the concept of conservation of angular momentum. The initial angular momentum (L_i) is zero since both the probe and rotor are at rest. The final angular momentum (L_f) can be expressed as:
L_f = I_r * ω_r
where ω_r is the angular velocity of the rotor.
Since angular momentum is conserved, we have:
L_i = L_f
0 = I_r * ω_r
Simplifying this expression, we can conclude that the angular velocity of the rotor is zero. This means that the rotor does not rotate.
Therefore, through the given information, we can determine that the rotor has not turned through any revolutions.
what is 6000*32.6 degrees?
momentum is conserved.
Iprobe*wprobe=Imotor*wmotor
INtegrate both sides over the same time
6000*wprobe*t=1*wm*t
6000*wp*t=wm*t
but wp*t=32.6
so the displacement of the motor is 6000*32.6 degrees