The wheels of a certain roller coaster are both
above and below the rails as shown in the
figure so that the car will not leave the rails.
The mass supported by this particular wheel
is 117 kg and the radius of this section of track
is 44 m.
The acceleration of gravity is 9.8 m/s
2
.
What is the magnitude of the force that the
track exerts on the wheel when the speed of
the car is 88 m/s?
Answer in units of N.
To find the magnitude of the force that the track exerts on the wheel, we can use the concept of centripetal force. The centripetal force is the force that keeps an object moving in a circular path.
The centripetal force can be determined using the formula:
F = (m * v^2) / r
where:
F is the centripetal force
m is the mass supported by the wheel (117 kg)
v is the speed of the car (88 m/s)
r is the radius of the track (44 m)
Substituting the given values into the formula:
F = (117 kg * (88 m/s)^2) / 44 m
Calculating this expression:
F = (117 kg * 7744 m^2/s^2) / 44 m
F = 2062152 kg·m/s^2 / 44 m
Simplifying the expression:
F = 46821 N
Therefore, the magnitude of the force that the track exerts on the wheel when the speed of the car is 88 m/s is 46821 N.
To find the magnitude of the force that the track exerts on the wheel, we can use Newton's second law of motion, which states that force is equal to mass multiplied by acceleration. In this case, the force exerted by the track on the wheel is the centripetal force needed to keep the car moving in a circular path.
The centripetal force can be calculated using the equation:
F = (m * v^2) / r
Where:
F is the centripetal force
m is the mass supported by the wheel
v is the speed of the car
r is the radius of the section of the track
Given:
m = 117 kg
v = 88 m/s
r = 44 m
Substituting the values into the equation, we have:
F = (117 kg * (88 m/s)^2) / 44 m
Calculating this expression, we get:
F = (117 kg * 7744 m^2/s^2) / 44 m
Simplifying further:
F = 2066688 kg*m/s^2 / 44 m
F = 46924 kg*m/s^2
Since the unit of force is Newtons (N), we need to convert the kg*m/s^2 to Newtons by using the relationship, 1 N = 1 kg*m/s^2. Therefore, the magnitude of the force that the track exerts on the wheel is:
F = 46924 N
So, the magnitude of the force is 46924 N.
Draw a free body diagram of the mass.
If we assume that the car is at the top of a loop, there should be a normal force N (force exerted by tracks on wheels) upward and mg downward.
The sum of the forces is also the total acceleration toward the center of the circle, so you get mvˆ2/r = N - mg
Then plug in the numbers and solve for N.
Source: my homework