How Does Centripetal Force Affect Roller Coaster Dynamics?

[dontcare]

A roller-caster vechicle has a mass of 500 kg when fully loaded with passengers (Fig p7.28) (a) If the vechile has a speed of 20.0 m/s at point A, what is the force of the track on the vehicle at this point? (b) What is the maximum speed the vehicle can have at point B in order for gravity to hold it on the track?

a) $$a_{c} = \frac{v^2}{r} = 40 m/s^2$$

 

[nrqed]

A roller-caster vechicle has a mass of 500 kg when fully loaded with passengers (Fig p7.28) (a) If the vechile has a speed of 20.0 m/s at point A, what is the force of the track on the vehicle at this point? (b) What is the maximum speed the vehicle can have at point B in order for gravity to hold it on the track?

a) $$a_{c} = \frac{v^2}{r} = 40 m/s^2$$

Draw a free body diagram. Identify all the forces. The acceleration will be toward the center of the circle (where is the object here? I am assuming that it`s at the very top of a loop? In that case you would have $a_y = - v^2/R$). Then apply Newton`s second law.

 

[kyle8921]

The force of the track on the vehicle at point A can be calculated using Newton's second law, F=ma, where F is the force, m is the mass, and a is the acceleration. In this case, the centripetal acceleration, a_{c}, is equal to the acceleration due to the force of the track. Therefore, the force of the track can be calculated as F=ma_{c} = 500 kg * 40 m/s^2 = 20,000 N.

b) In order for gravity to hold the vehicle on the track at point B, the centripetal force must be equal to the force of gravity acting on the vehicle. This can be calculated using the formula F=mg, where m is the mass of the vehicle and g is the acceleration due to gravity. Therefore, the maximum speed the vehicle can have at point B can be calculated as v=\sqrt{\frac{gr}{m}} = \sqrt{\frac{9.8 m/s^2 * 50 m}{500 kg}} = 10 m/s.

It is important to note that this calculation assumes ideal conditions and does not take into account any external factors such as friction or air resistance. It is always important to consider these factors in real-world scenarios to ensure the safety and accuracy of calculations.