A roller-coaster vehicle has a mass of 400 kg when fully loaded with passengers. r1 = 11 m and r2 = 17 m. If the vehicle has a speed of 17 m/s at A, what is the magnitude of the force that the track exerts on the vehicle at this point?

To find the magnitude of the force that the track exerts on the vehicle at point A, we can use the principle of conservation of mechanical energy.

The mechanical energy of the roller-coaster vehicle is the sum of its kinetic energy and gravitational potential energy at point A. Mathematically, we can express this as:

E1 = KE1 + PE1

Where:
E1 is the total mechanical energy at point A
KE1 is the kinetic energy at point A
PE1 is the gravitational potential energy at point A

First, let's calculate the kinetic energy at point A.

The kinetic energy (KE) of an object is given by the formula:

KE = (1/2) * m * v^2

Where:
m is the mass of the object
v is the velocity of the object

In this case, the mass of the roller-coaster vehicle is 400 kg, and the velocity at point A is 17 m/s. Substituting these values into the formula, we can calculate the kinetic energy at point A.

KE1 = (1/2) * 400 kg * (17 m/s)^2
KE1 = 115,600 J

Next, let's calculate the gravitational potential energy at point A.

The gravitational potential energy (PE) of an object near the surface of the Earth is given by the formula:

PE = m * g * h

Where:
m is the mass of the object
g is the acceleration due to gravity (approximately 9.8 m/s^2 near the surface of the Earth)
h is the height above a reference point

In this case, the reference point is not mentioned, so we'll assume it is the height of the roller-coaster track at point A. The height can be calculated as the difference between the radius of the track at point A (r1) and the radius of the Earth (which we'll assume as negligible).

h = r1

Substituting the values into the formula, we can calculate the gravitational potential energy at point A.

PE1 = 400 kg * 9.8 m/s^2 * 11 m
PE1 = 43,120 J

Now that we have the kinetic energy and gravitational potential energy at point A, we can calculate the total mechanical energy at point A by summing them.

E1 = KE1 + PE1
E1 = 115,600 J + 43,120 J
E1 = 158,720 J

According to the conservation of mechanical energy, the total mechanical energy should be constant along the roller-coaster track. So, the total mechanical energy at any other point (let's say point B) should also be 158,720 J.

At point B, the roller-coaster vehicle is located on a different radius of the track (r2 = 17 m). We'll assume that the roller-coaster vehicle is frictionless and there is no air resistance, so there is no energy loss along the track.

At point B, the roller-coaster vehicle has gravitational potential energy and kinetic energy as well. Using the same formulas as before, with the new radius (r2 = 17 m), we can find the total mechanical energy at point B.

KE2 = (1/2) * 400 kg * (v^2)
PE2 = 400 kg * 9.8 m/s^2 * 17 m
E2 = KE2 + PE2

Since the total mechanical energy is conserved, E2 = E1 = 158,720 J. Now, we can solve for the velocity (v) at point B using the given information.

158,720 J = (1/2) * 400 kg * (v^2) + 400 kg * 9.8 m/s^2 * 17 m

Simplifying the equation, we can find the value of v.

Now, the problem is asking for the magnitude of the force that the track exerts on the vehicle at point A. At any point on a curved track, the net force acting on the vehicle is equal to the centripetal force, which is directed towards the center of the track. The magnitude of the centripetal force can be calculated using the formula:

F = m * (v^2) / r

Where:
F is the magnitude of the force
m is the mass of the vehicle (400 kg)
v is the velocity of the vehicle
r is the radius of the track (r1 = 11 m at point A)

Substituting the values into the formula, we can calculate the magnitude of the force.

F = 400 kg * (v^2) / 11 m

After calculating the value of v from the previous equation and substituting it into this formula, you will find the magnitude of the force that the track exerts on the vehicle at point A.

To find the magnitude of the force that the track exerts on the vehicle at point A, we can use the principles of circular motion.

The force exerted by the track on the vehicle at point A is the centripetal force that keeps the vehicle moving in a circle.

The centripetal force can be calculated using the equation:

F = (m * v^2) / r

where:
F is the centripetal force,
m is the mass of the vehicle,
v is the velocity of the vehicle, and
r is the radius of the circular path.

Given:
m = 400 kg (mass of the roller-coaster vehicle)
v = 17 m/s (speed of the vehicle at point A)
r1 = 11 m (radius of the circular path at point A)

Using the given values, we can substitute them into the equation to find the force:

F = (400 kg * (17 m/s)^2) / 11 m

Calculating this equation gives:

F = 400 kg * (289 m^2/s^2) / 11 m
F = 115600 kg m^2/s^2 / 11 m
F = 10509.1 N

Therefore, the magnitude of the force that the track exerts on the vehicle at point A is 10509.1 Newtons.