A child exerts a tangential 41.9-N force on the rim of a disk-shaped merry-go-round with a radius of 2.25 m. If the merry-go-round starts at rest and acquires an angular speed of 0.086 rev/s in 3.57 s, what is its mass?
To solve this problem, we can use the concept of torque and angular acceleration.
The torque exerted on an object is defined as the product of the force applied and the lever arm distance. In this case, the force applied by the child is tangential to the merry-go-round, so the lever arm distance is equal to the radius of the merry-go-round.
The torque (τ) can be calculated using the formula:
τ = r * F
Where:
τ is the torque
r is the radius of the merry-go-round
F is the force applied
Since the object starts at rest and acquires an angular speed, we can use the equation:
τ = I * α
Where:
τ is the torque
I is the moment of inertia of the object
α is the angular acceleration
In this problem, we are given the radius (r), the applied force (F), the angular speed (ω), and the time (t). We need to find the moment of inertia (I).
First, let's calculate the torque (τ) using the given force (F) and radius (r):
τ = r * F
τ = 2.25 m * 41.9 N
τ = 94.275 Nm
Next, let's calculate the angular acceleration (α) using the given angular speed (ω) and time (t):
α = Δω / Δt
α = (0.086 rev/s) / (3.57 s)
α = 0.0241 rev/s^2
Now, we can rearrange the equation τ = I * α to solve for the moment of inertia (I):
I = τ / α
I = 94.275 Nm / 0.0241 rev/s^2
I = 3908.921 kg·m^2
Finally, the mass (m) of the merry-go-round can be calculated using the formula:
I = m * r^2
Rearranging the equation, we get:
m = I / r^2
m = 3908.921 kg·m^2 / (2.25 m)^2
m = 762.543 kg
Therefore, the mass of the merry-go-round is approximately 762.543 kg.
To find the mass of the merry-go-round, we can use the principle of conservation of angular momentum.
The formula for angular momentum is L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
In this case, the merry-go-round starts from rest, so its initial angular momentum is zero. After the child exerts the tangential force, the merry-go-round acquires an angular speed of 0.086 rev/s.
To calculate the moment of inertia, we can use the formula I = MR^2, where M is the mass and R is the radius of the merry-go-round.
Since we are looking for the mass of the merry-go-round, we can rearrange the formula to M = I / R^2.
First, let's convert the angular speed from rev/s to rad/s. One revolution is equal to 2π radians, so 0.086 rev/s is equal to 0.086 * 2π rad/s.
Substituting the given values into the equations:
L = Iω
0 = I * 0.086 * 2π
Simplifying this equation:
0 = I * 0.172π
Now we can solve for the moment of inertia:
I = 0 / 0.172π
I = 0
Since the moment of inertia is zero, this means that the mass of the merry-go-round is also zero.
Therefore the mass of the merry-go-round cannot be determined given the information provided in the question.