A roller coaster heads into a circular loop with radius 5m.
At what minimum speed must the coaster go at the top of the loop to be sure it stays on the track in m/s?
To determine the minimum speed at the top of the loop, we need to consider the forces acting on the roller coaster.
At the top of the loop, two forces are acting on the roller coaster:
1. Gravity, which acts vertically downwards.
2. The normal force, which acts towards the center of the loop.
To ensure that the roller coaster stays on the track, the normal force must be greater than or equal to zero. This means that the net force acting towards the center of the loop must be greater than or equal to zero.
The net force at the top of the loop is given by the difference between the gravitational force and the centripetal force acting towards the center of the loop.
The gravitational force is equal to the weight of the roller coaster, which is given by the mass of the roller coaster (m) multiplied by the acceleration due to gravity (g). Assuming a mass of 1000 kg and a gravitational acceleration of 9.8 m/s^2, the weight of the roller coaster is:
Weight = m * g = 1000 kg * 9.8 m/s^2 = 9800 N.
The centripetal force is given by:
Centripetal force = m * v^2 / r,
where m is the mass of the roller coaster, v is its velocity, and r is the radius of the loop. In this case, m = 1000 kg and r = 5 m.
Substituting these values into the equation for the net force:
Net force = Weight - Centripetal force
= 9800 N - (1000 kg * v^2 / 5 m)
= 9800 N - 2000 v^2 N/m.
For the roller coaster to stay on the track, the net force must be greater than or equal to zero:
Net force ≥ 0.
Substituting the equation for the net force:
9800 N - 2000 v^2 N/m ≥ 0.
Simplifying the inequality:
2000 v^2 ≤ 9800 N.
Dividing both sides by 2000:
v^2 ≤ 4.9 m^2/s^2.
Taking the square root of both sides:
v ≤ 2.2 m/s.
Therefore, the roller coaster must have a minimum speed of 2.2 m/s at the top of the loop to ensure it stays on the track.