A merry-go-round of mass 1640kg has a radius of 7.5m. How much work is required to accelerate it from rest to a rotation rate of 1 rev. per 8 seconds? (assume a solid cylinder)

Question

A merry-go-round of mass 1640kg has a radius of 7.5m. How much work is required to accelerate it from rest to a rotation rate of 1 rev. per 8 seconds? (assume a solid cylinder)

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Answer 1

To determine the amount of work required to accelerate the merry-go-round, we need to consider the rotational kinetic energy. The formula for rotational kinetic energy is:

KE = (1/2) I ω^2

where KE is the rotational kinetic energy, I is the moment of inertia, and ω is the angular velocity.

The moment of inertia for a solid cylinder can be calculated using the formula:

I = (1/2) m r^2

where m is the mass of the merry-go-round and r is the radius.

First, let's calculate the moment of inertia:

I = (1/2) m r^2
= (1/2) (1640 kg) (7.5 m)^2

I = 1.8595 × 10^5 kg·m^2

Next, we need to find the angular velocity (ω) in radians per second. The relationship between angular velocity (in radians per second) and the number of revolutions per second (rev/s) is:

ω = 2π × (rev/s)

Given that the rotation rate is 1 rev per 8 seconds, we can calculate ω:

ω = 2π × (1/8 s)
= π/4 rad/s

Now we can calculate the rotational kinetic energy (KE):

KE = (1/2) I ω^2
= (1/2) (1.8595 × 10^5 kg·m^2) (π/4 rad/s)^2

KE = 2.9027 × 10^4 Joules

Therefore, the amount of work required to accelerate the merry-go-round from rest to a rotation rate of 1 rev per 8 seconds is approximately 2.9027 × 10^4 Joules.

Answer 2

To find the work required to accelerate the merry-go-round, we need to determine the rotational kinetic energy of the system at a rotation rate of 1 revolution per 8 seconds.

The rotational kinetic energy of a solid cylinder can be given by the formula:

KE = (1/2) * I * ω^2

where:
KE is the rotational kinetic energy,
I is the moment of inertia,
ω is the angular velocity (in rad/s).

The moment of inertia of a solid cylinder can be calculated using the formula:

I = 0.5 * m * r^2

where:
m is the mass of the cylinder,
r is the radius of the cylinder.

Given:
Mass of the merry-go-round (m) = 1640 kg
Radius of the merry-go-round (r) = 7.5 m
Rotation rate (ω) = 1 revolution / 8 seconds = 1 rev / 8 s

First, let's convert the rotation rate to rad/s:
1 revolution = 2π radians
1 second = 8 seconds

So, the rotation rate (ω) = (2π rad) / (8 s)

Now, let's calculate the moment of inertia (I):
I = 0.5 * m * r^2
= 0.5 * 1640 kg * (7.5 m)^2

Finally, we can calculate the rotational kinetic energy (KE):
KE = (1/2) * I * ω^2

Let's plug the values into the formula and solve for KE.