The radius of curvature of a loop-the-loop for a roller coaster is 11.8 m. At the top of the loop (with the car inside the loop), the force that the seat exerts on a passenger of mass m is 0.80mg. Find the speed of the roller coaster at the top of the loop.

m/s

To find the speed of the roller coaster at the top of the loop, we can use the concept of centripetal force.

The force exerted by the seat on the passenger at the top of the loop is the net force acting towards the center of the loop. Therefore, the centripetal force is given by:
Fc = 0.80mg

The centripetal force is given by the equation:
Fc = (mv^2) / r

Where:
Fc = centripetal force
m = mass of the roller coaster car and the passenger
v = velocity of the roller coaster at the top of the loop
r = radius of curvature

Combining the two equations, we have:
0.80mg = (mv^2) / r

Canceling out the mass (m) on both sides of the equation, we get:
0.80g = v^2 / r

Multiplying both sides of the equation by r, we get:
0.80gr = v^2

Finally, taking the square root of both sides of the equation, we can solve for v:
v = sqrt(0.80gr)

Using the given radius of curvature (r = 11.8 m) and assuming the gravitational acceleration (g = 9.8 m/s^2), we can calculate the speed of the roller coaster at the top of the loop:
v = sqrt(0.80 * 9.8 * 11.8) ≈ 9.79 m/s

Therefore, the speed of the roller coaster at the top of the loop is approximately 9.79 m/s.

To find the speed of the roller coaster at the top of the loop, we can use the concept of centripetal force.

1. First, we need to identify the forces acting on the passenger at the top of the loop.
- The force of gravity acting on the passenger, given by mg (where m is the mass of the passenger and g is the acceleration due to gravity).
- The normal force exerted by the seat on the passenger, given by F_n (which is equal to 0.80mg, according to the given information).
- The centripetal force required to keep the passenger moving in a circular path, which is provided by the net force at the top of the loop.

2. Next, let's determine the net force acting on the passenger at the top of the loop.
- The net force is the difference between the gravitational force and the normal force, both of which act downwards at the top of the loop (considering upward direction as positive).
- Net force = mg - F_n

3. The net force is also equal to the centripetal force, which can be calculated using the following formula:
- Centripetal force = (mass x velocity^2) / radius of curvature

4. Equating the net force and the centripetal force, we can write:
- mg - F_n = (m x v^2) / r, where v is the speed of the roller coaster at the top of the loop.

5. Rearranging the equation to solve for v, we have:
- v^2 = (r x (g - F_n)) / m

6. Taking the square root of both sides, we get:
- v = sqrt((r x (g - F_n)) / m)

7. Now we can substitute the given values into the equation:
- r = 11.8 m, g = 9.8 m/s^2 (acceleration due to gravity), and F_n = 0.80mg.

v = sqrt((11.8 x (9.8 - 0.80mg)) / m)

8. Finally, we can simplify the equation by substituting F_n = 0.80mg:
- v = sqrt((11.8 x (9.8 - 0.80mg)) / m)

Now, you can substitute the value of m (mass of the passenger) into the equation to find the speed of the roller coaster at the top of the loop in m/s.