A new ride is being designed for Dubailand. The ride will have 80 identical carts placed 15 m away from the center. Each cart has a mass of 250 kg. If the moment of inertia of each connecting rod system is 35 kg m² and the ride spins with an angular speed of 0.21 rad/s, what is the total kinetic energy of the ride during a test run with empty carts?
The total moment of inertia of the ride can be calculated as follows:
I = 80(35 kg m²) + 80(250 kg)(15 m)²
I = 807,500 kg m²
The kinetic energy of the ride can be calculated using the formula:
K = (1/2)Iω²
where ω is the angular speed in radians per second.
Plugging in the values, we get:
K = (1/2)(807,500 kg m²)(0.21 rad/s)²
K = 15,585.53 J
Therefore, the total kinetic energy of the ride during a test run with empty carts is 15,585.53 J.
To find the total kinetic energy of the ride during a test run with empty carts, we need to calculate the rotational kinetic energy of each cart and then sum it up for all the carts.
Rotational Kinetic Energy (KEₙ) can be calculated using the formula: KEₙ = (1/2) * I * ω², where I is the moment of inertia and ω is the angular speed.
Given:
- Number of carts (n) = 80
- Distance from the center (r) = 15 m
- Mass of each cart (m) = 250 kg
- Moment of inertia of each connecting rod system (I) = 35 kg m²
- Angular speed (ω) = 0.21 rad/s
Step 1: Calculate the moment of inertia of each cart (Ic)
The moment of inertia about the center of each cart can be calculated using the parallel axis theorem. The moment of inertia of a point mass about an axis is given by I = m * r², where m is the mass and r is the distance from the axis of rotation.
The distance from the center to each cart is the same, r = 15 m.
Ic = m * r²
Ic = 250 kg * (15 m)²
Ic = 250 kg * 225 m²
Ic = 56250 kg m²
Step 2: Calculate the rotational kinetic energy for each cart (KEc)
Using the formula KEc = (1/2) * Ic * ω²
KEc = (1/2) * 56250 kg m² * (0.21 rad/s)²
KEc = 0.5 * 56250 kg m² * 0.0441 rad²/s²
KEc = 1237.5 kg m²/s² * 0.0441 rad²/s²
KEc = 54.41375 kg m²/s²
Step 3: Calculate the total kinetic energy of the ride (KE)
The total kinetic energy (KE) is the sum of the rotational kinetic energy for each cart.
KE = n * KEc
KE = 80 * 54.41375 kg m²/s²
KE = 4353.10 kg m²/s²
Therefore, the total kinetic energy of the ride during a test run with empty carts is approximately 4353.10 kg m²/s².