Roller Coaster Cart Mass = 200kg
String Compressed by 5m
Initial Velocity 25 m/s
PE = 2,500 J
KE = 62,500 J
Use the law of conservation of mechanical energy to find the maximum radius the loop can have before the cart falls off
To find the maximum radius of the loop before the cart falls off, we can use the law of conservation of mechanical energy. According to this law, the total mechanical energy of an object remains constant as long as no external forces are acting on it.
The total mechanical energy of the roller coaster cart at the initial point is the sum of its potential energy (PE) and kinetic energy (KE), which is given as 2,500 J and 62,500 J respectively. This means that the initial total mechanical energy is 65,000 J.
At the highest point of the loop, all of the initial kinetic energy is converted into potential energy, where the cart has zero velocity. Therefore, at the top of the loop, the total mechanical energy is equal to the potential energy.
Using the equation for potential energy, PE = m * g * h, where m is the mass of the cart, g is the acceleration due to gravity, and h is the height, we can solve for the height at the top of the loop.
PE = m * g * h
2,500 J = 200 kg * 9.8 m/s^2 * h
Solving for h, we find:
h = 2,500 J / (200 kg * 9.8 m/s^2)
h = 1.28 m
Now, we can use the radius of the loop to determine the height using the equation h = R - R * cos(theta), where R is the radius of the loop and theta is the angle between the vertical and the string.
Given that the string is compressed by 5 m, the height of the cart at the highest point of the loop would be the compressed length of the string plus the height at the top of the loop. Therefore:
h = 1.28 m + 5 m
h = 6.28 m
Now, we can substitute this value into the equation for the potential energy to find the maximum radius of the loop:
PE = m * g * h
2,500 J = 200 kg * 9.8 m/s^2 * (6.28 m - R * cos(theta))
Since we do not know the angle, cos(theta) is equal to 1 at the highest point of the loop, and the equation simplifies to:
2,500 J = 200 kg * 9.8 m/s^2 * (6.28 m - R)
Now, we can solve for the maximum radius:
R = (200 kg * 9.8 m/s^2 * 6.28 m - 2,500 J) / (200 kg * 9.8 m/s^2)
R ≈ 3.95 m
Therefore, the maximum radius the loop can have before the cart falls off is approximately 3.95 meters.