A roller coaster cart with a mass of 13 kg travels around two frictionless, circular loops. It just barely makes it around the top of the first circular loop which has a radius of 24 meters. It then travels around the other loop with a radius of 14 meters. What is the normal force at the top of the second loop in Newtons?
To find the normal force at the top of the second loop, we can start by analyzing the forces acting on the roller coaster cart at that point. At the top of the loop, there are two forces acting on the cart: the gravitational force (weight) and the normal force. Since the roller coaster cart is just barely making it around the loop, it means that the normal force is equal to zero at this point. This is the critical point where the cart is on the verge of losing contact with the track.
To calculate the normal force at the top of the second loop, we can use the concept of centripetal force. At the top of the loop, the only horizontal force acting on the cart is the gravitational force. The centripetal force is provided solely by the normal force.
The centripetal force is given by the formula:
Fc = m * v² / r
Where Fc is the centripetal force, m is the mass of the cart, v is the velocity, and r is the radius of the loop.
Since the cart is just barely making it around the loop, we know that the centripetal force (Fc) equals the gravitational force (mg).
Therefore, mg = m * v² / r
Rearranging the equation to solve for v², we get:
v² = g * r
Where g is the acceleration due to gravity (approximately 9.8 m/s²).
Now we can calculate the velocity (v) at the top of the loop using the radius (r) of the second loop:
v² = 9.8 m/s² * 14 m
v² = 137.2 m²/s²
Taking the square root of both sides, we find:
v ≈ 11.71 m/s
Now that we know the velocity at the top of the loop, we can calculate the normal force (N) using the equation:
N = m * ( g + v² / r )
Substituting the given values:
N = 13 kg * (9.8 m/s² + 11.71 m/s) / 14 m
Calculating further:
N ≈ 90.46 N
Therefore, the normal force at the top of the second loop is approximately 90.46 Newtons.