A circular disk of mass 0.4 kg and radius 36 cm, initially not rotating, slips down a thin spindle onto a turntable (disk) of mass 1.7 kg and the same radius, rotating freely at 3.4
a) Find the new angular velocity of the combination;
b) The change in the kinetic energy?
c) If the motor is switched on after the disk has landed, what is the constant torque needed to regain the original speed in 2.2 s?
I figured out a) and b), but I'm not sure what c) is asking or how to solve it.
3.4 WHAT??
I assume radians/second
but hard to say. I will use that but if RPS or something, convert
I disk = (1/2)m r^2
so
Id = .5*.4*.36^2
and
It = .5 * 1.7 * .36^2
initial angular momentum = It*3.4
final angular momentum (the same of course)
= It omega +Id omega = omega(It+Id)
so
It * 3.4 = (It+Id)omega
but It =1.7/.4 * Id
so It = 4.25 Id
so
3.4 (4.25 Id) = (5.25 Id) omega
omega = 3.4 (4.25/5.25)
omega = 2.75 whatever your units are
----------------
initial Ke = .5 It(3.4)^2
final Ke = .5 (It+Id)(2.75)^2
-------------------
Torque = moment = change in I*omega /time
[just like force =change in m*v/t]
= (It+Id) (3.4-2.75) / 2.2
Thanks Damon you're the real MVP!
It was rad/s sorry.
To answer part c), we need to calculate the constant torque needed to regain the original speed in 2.2 seconds after the motor is switched on.
First, let's review the concepts involved. Torque is defined as the rate of change of angular momentum, which can be given by the equation:
Torque = Change in angular momentum / Time
Angular momentum can be calculated as the product of moment of inertia and angular velocity. Since the moment of inertia of a disk is given by the equation:
I = (1/2) * mass * radius^2
We can express the initial angular momentum as:
L1 = I1 * 0 (since the initial angular velocity is zero)
Similarly, the final angular momentum can be expressed as:
L2 = I2 * ω2 (where ω2 is the final angular velocity)
Now, the change in angular momentum is given by:
ΔL = L2 - L1
Substituting the values of L1, L2, and moment of inertia equations, we get:
ΔL = (1/2 * mass1 * radius1^2 * ω2) - 0
Now, the torque required to produce this change in angular momentum over a specific time interval can be given by the equation:
Torque = ΔL / Time
Substituting the value of ΔL, we get:
Torque = (1/2 * mass1 * radius1^2 * ω2) / Time
Plugging in the values from the given information, we have:
mass1 = 0.4 kg
radius1 = 36 cm = 0.36 m
ω2 = the angular velocity calculated in part a)
Time = 2.2 seconds
Now, you can calculate the constant torque needed to regain the original speed by substituting these values into the equation mentioned above.