A 190-kg merry-go-round in the shape of a uniform, solid, horizontal disk of radius 1.50 m is set in motion by wrapping a rope about the rim of the disk and pulling on the rope. What constant force would have to be exerted on the rope to bring the merry-go-round from rest to an angular speed of 0.700 rev/s in 2.00 s? (State the magnitude of the force.)
torque=force*radius=I*alpha
force=(momentinertiadisk)alpha/radius
alpha=.700*2PI/2
solve for force.
1.1
To find the force required to bring the merry-go-round from rest to the given angular speed, we need to apply the principles of rotational motion and Newton's second law.
First, we can calculate the moment of inertia of the merry-go-round, which depends on its mass and radius. The moment of inertia for a solid disk rotating about its center is given by the formula:
I = (1/2) * m * r^2
where I is the moment of inertia, m is the mass, and r is the radius.
Plugging in the values, we have:
I = (1/2) * 190 kg * (1.50 m)^2
I = 213.75 kg * m^2
Next, we can determine the change in angular speed (ω) using the formula:
ω = (final angular speed - initial angular speed) / time
Plugging in the values we have:
ω = (0.700 rev/s - 0 rev/s) / 2.00 s
ω = 0.350 rev/s
Now, we can use Newton's second law for rotational motion, which states that the torque (τ) is equal to the moment of inertia (I) multiplied by the angular acceleration (α):
τ = I * α
The torque is given by the force (F) multiplied by the radius (r):
τ = F * r
Therefore, we can rewrite the equation as:
F * r = I * α
Since we are given the initial angular speed (ω), final angular speed (0.700 rev/s), and the time (2.00 s), we can calculate the angular acceleration (α) using the formula:
α = (final angular speed - initial angular speed) / time
α = (0.700 rev/s - 0 rev/s) / 2.00 s
α = 0.350 rev/s^2
Substituting this into the equation, we have:
F * r = I * α
F * 1.50 m = 213.75 kg * m^2 * 0.350 rev/s^2
Simplifying:
F = (213.75 kg * m^2 * 0.350 rev/s^2) / 1.50 m
F = 499.25 N
Therefore, the magnitude of the force required to bring the merry-go-round from rest to an angular speed of 0.700 rev/s in 2.00 s is 499.25 N.