A child pushes tangentially on a small hand driven merry- go- round. She is able to accelerate it from rest to 16 rpm in 10 s. Modeling the merry go round as a uniform disk of radius 4.0 m and mass 770 kg, find the torque required to produce that acceleration. Neglect the frictional torque.
90000 N
the principle here is simple:
torque=moment*acceleration
where acceleration is found by wfinal^2=1/2 acceleation*time
wfinal=16*2PI/60 rad/sec
use your table of moments to find the disk moment.
To find the torque required to produce the given acceleration, we can use the equation:
Torque = Moment of inertia * Angular acceleration
First, let's determine the moment of inertia of the merry-go-round. The moment of inertia of a uniform disk can be calculated using the formula:
I = (1/2) * m * r^2
where I is the moment of inertia, m is the mass of the object, and r is the radius of the object.
Given that the mass of the merry-go-round is 770 kg and the radius is 4.0 m, we can substitute these values into the formula to find the moment of inertia:
I = (1/2) * 770 kg * (4.0 m)^2
I = 3080 kg·m^2
Next, let's determine the angular acceleration of the merry-go-round. We are given that it goes from rest to 16 rpm in 10 seconds. To convert rpm to rad/s, we can use the formula:
Angular acceleration = (Angular velocity_final - Angular velocity_initial) / time
First, let's convert the given final angular velocity from rpm to rad/s:
Angular velocity_final = 16 rpm * (2π rad/1 min) * (1 min/60 s)
Angular velocity_final = 16 * (2π/60) rad/s
Angular velocity_final = 33.51 rad/s
Since the merry-go-round starts from rest (angular velocity_initial = 0 rad/s), we can substitute the values into the formula to determine the angular acceleration:
Angular acceleration = (33.51 rad/s - 0 rad/s) / 10 s
Angular acceleration = 3.351 rad/s^2
Now, we can plug the obtained values of moment of inertia and angular acceleration into the equation to find the torque:
Torque = 3080 kg·m^2 * 3.351 rad/s^2
Torque = 10323.18 N·m
Therefore, the torque required to produce the given acceleration is approximately 10323.18 N·m.