Suppose the vertical loop has a radius of 8.92 m. What is the apparent weight (Wap) of a rider on the roller coaster at the bottom of the loop? (Assume that friction between roller coaster and rails can be neglected. Give your answer in terms of m and g.)
Didn't you ask this yesterday? As BobPursley answered then, you need to know the velocity at the bottom of the loop. You still have not provided that information.
You can make a calculation of that velocity by using conservation of energy IF the velocity is zero at the top of the loop. They may expect you to make that assumption.
V(@ bottom)= sqrt(2gH) = 2 sqrt(gR)
Wap (@bottom)
= M (V^2/R + g)
= 5 M g
It is somewhat surprising that it is independent of R.
I have seen rides like that at carnival amusement parks, but never went on one.
so Wap will equal in terms of m and g this.
m ( (sqrt ( r*g )/ r ) + g )
v^2 = 4 g r
so
v^2/r = 4 g
so weight = m(4g+g) = 5 m g
To calculate the apparent weight (Wap) of a rider on a roller coaster at the bottom of the loop, you need to consider the forces acting on the rider.
At the bottom of the loop, the rider experiences two forces: the gravitational force (mg) and the normal force (N). The normal force is the force exerted by the track on the rider, perpendicular to the track.
The apparent weight (Wap) is the net force acting on the rider, which can be calculated as the difference between the gravitational force and the normal force:
Wap = mg - N
Since the rider is at the bottom of the loop, the normal force is directed upwards and the gravitational force is directed downwards. In this scenario, the normal force is greater than the gravitational force. Therefore, the apparent weight is negative, indicating that the rider feels lighter than their actual weight.
To find the normal force, you can use the centripetal force equation:
N = m * (v^2 / r)
Where:
m is the mass of the rider
v is the velocity of the rider at the bottom of the loop
r is the radius of the loop
In this case, the radius of the loop is given as 8.92 m. However, the velocity of the rider at the bottom of the loop is not provided in the question. To proceed, you need to calculate the velocity using the principles of circular motion.
The centripetal force required for circular motion is provided by the net force acting towards the center of the loop. In this case, at the bottom of the loop, the net force is equal to the difference between the normal force and the gravitational force:
Net Force = N - mg
The net force can also be written as the product of mass (m) and centripetal acceleration (ac):
Net Force = m * ac
Therefore, you have:
N - mg = m * ac
The centripetal acceleration (ac) can be calculated using the following formula:
ac = v^2 / r
Rearranging the equation, you get:
v^2 = ac * r
Substituting this value for v^2 in the previous equation:
N - mg = m * (v^2 / r)
Simplifying:
N - mg = m * (ac * r) / r
N - mg = m * ac
N = m * (ac + g)
Where g is the acceleration due to gravity.
Now, you know the velocity for the given radius is 8.92 m and the acceleration due to gravity is approximately 9.8 m/s^2. Substituting these values into the equation, you can calculate the normal force.
Finally, you can calculate the apparent weight:
Wap = mg - N
Given m and g, you can now substitute the values into the equation and solve for Wap.