Final answer:
The minimum speed at which a roller coaster must travel when upside down at the top of a 7.4 m radius loop to keep passengers from falling out is 8.5 m/s. This is calculated by setting the centripetal force equal to the gravitational force when the normal force is zero.
Step-by-step explanation:
To determine the minimum speed at which a roller coaster must travel when upside down at the top of a loop to ensure passengers do not fall out, we must consider the condition where the centripetal force provided by the circular motion equals the gravitational force acting on the passengers. This scenario occurs when the normal force (the force from the coaster car on the passenger) is zero, as that represents the minimum condition for the passengers to stay in their seats without falling out.
At the top of a loop, the force of gravity and the centripetal force must be at least equal for this condition to be met. The formula to calculate centripetal force is F = m * v^2 / r, where m is the mass, v is the speed, and r is the radius. Setting this equal to the gravitational force, m * g (where g is the acceleration due to gravity), and solving for v gives us the minimum speed for safe traversal of the loop. For a 7.4 m radius loop, this calculation would be as follows:
v = √(r * g)
Substituting the known values (g = 9.8 m/s^2, r = 7.4 m), we get:
v = √(7.4 m * 9.8 m/s^2) = √(72.52) = 8.5 m/s
Therefore, the roller coaster must travel at a minimum speed of 8.5 m/s at the top of the loop to ensure that the passengers will not fall out.