A child pushes tangentially on a small hand driven merry- go- round. She is able to accelerate it from rest to 18 rpm in 10 s. Modeling the merry go round as a uniform disk of radius 2.0 m and mass 800 kg, find the torque required to produce that acceleration. Neglect the frictional torque.
To find the torque required to produce the acceleration of the merry-go-round, we can use the following steps:
Step 1: Find the angular acceleration
The angular acceleration (α) can be found by converting the final angular velocity (ωf) from rpm to rad/s and dividing it by the time (t) taken:
ωf = 18 rpm = 18 × 2π rad/60 s = 1.88 rad/s
t = 10 s
α = ωf / t = 1.88 rad/s / 10 s = 0.188 rad/s²
Step 2: Find the moment of inertia
The moment of inertia (I) for a uniform disk rotating about its central axis is given by:
I = (1/2) × m × r²
where m is the mass and r is the radius.
m = 800 kg
r = 2.0 m
I = (1/2) × 800 kg × (2.0 m)² = 1600 kg·m²
Step 3: Calculate the torque
The torque (τ) required to produce the angular acceleration is given by Newton's second law for rotational motion:
τ = I × α
τ = 1600 kg·m² × 0.188 rad/s² = 300.8 N·m
Therefore, the torque required to produce the acceleration is 300.8 N·m.
To find the torque required to produce the acceleration of the merry-go-round, we can use the equation:
Torque = Moment of Inertia * Angular Acceleration
First, let's find the moment of inertia of the merry-go-round. We are given that it is a uniform disk, so the moment of inertia can be calculated using the formula:
Moment of Inertia (I) = (1/2) * Mass * Radius^2
Given values:
Mass (m) = 800 kg
Radius (r) = 2.0 m
Substituting the values into the formula, we get:
I = (1/2) * 800 kg * (2.0 m)^2
I = 1600 kgm^2
Now, let's find the angular acceleration (α) of the merry-go-round. We are told that it went from rest to 18 rpm (revolutions per minute) in 10 seconds. We need to convert the final angular velocity to radians per second:
Angular Velocity = (18 rpm) * (2π rad/rev) * (1 min/60 s)
Angular Velocity = 3π rad/s
Using the equation for angular acceleration (α):
Angular Acceleration = (Final Angular Velocity - Initial Angular Velocity) / Time
Given values:
Initial Angular Velocity = 0 rad/s
Final Angular Velocity = 3π rad/s
Time (t) = 10 s
Substituting the values into the formula, we get:
Angular Acceleration = (3π rad/s - 0 rad/s) / 10 s
Angular Acceleration = (3π rad/s) / 10 s
Angular Acceleration = (3/10)π rad/s^2
Finally, we can calculate the torque (τ) required using the equation:
Torque = Moment of Inertia * Angular Acceleration
Substituting the values, we get:
Torque = 1600 kgm^2 * (3/10)π rad/s^2
Torque ≈ 1507.96 Nm
Therefore, the torque required to produce the acceleration of the merry-go-round is approximately 1507.96 Nm.