Passengers riding in the Great Six Flags Air Racer are spun around a tall steel tower (see the figure). At top speed the planes fly at a 56.0 o bank approximately 12.0 m from the tower. In this position the support chains make an angle of 56.0 o with the vertical. Calculate the speed of the planes.
(There is a figure here)
To calculate the speed of the planes, we can use the concept of centripetal force and the relationship between speed, radius, and bank angle.
Here's how you can find the speed of the planes:
1. Identify the relevant information from the problem:
- Bank angle (θ) = 56.0 °
- Radius (r) = 12.0 m
2. Understand the relationship between speed, radius, and bank angle:
- The relationship is given by: tan(θ) = v^2 / (g * r), where v is the speed of the planes and g is the acceleration due to gravity (approximately 9.8 m/s^2).
- Rearranging the equation, we get: v = √(tan(θ) * g * r)
3. Substitute the values into the equation:
- tan(56.0 °) ≈ 1.477 (you can use a calculator to find this value)
- g ≈ 9.8 m/s^2
- r = 12.0 m
Plugging these values into the equation, we get:
v = √(1.477 * 9.8 * 12.0)
4. Calculate the speed of the planes:
- Calculate the expression inside the square root: 1.477 * 9.8 * 12.0 ≈ 170.728
- Taking the square root of 170.728, we find that v ≈ 13.07 m/s (rounded to two decimal places)
Therefore, the speed of the planes is approximately 13.07 m/s.
To find the speed of the planes, we can use the concept of centripetal force. The force of gravity acting on the planes is balanced by the tension in the support chains.
1. First, let's find the tension in the support chains. The vertical component of tension provides the force that counteracts gravity. Since the support chains make an angle of 56.0° with the vertical, the vertical component of tension is given by:
T * cos(56.0°) = mg --(Equation 1)
where T is the tension in the support chains, m is the mass of the planes, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Now, let's find the horizontal component of tension. This component provides the centripetal force required to keep the planes moving in a circular path. The horizontal component of tension is given by:
T * sin(56.0°) = (mv^2) / r --(Equation 2)
where v is the speed of the planes and r is the distance from the tower to the planes (12.0 m).
3. Next, let's solve Equation 1 for T:
T = (mg) / cos(56.0°)
4. Substitute the expression for T from Equation 3 into Equation 2:
[(mg) / cos(56.0°)] * sin(56.0°) = (mv^2) / r
5. Simplify the equation:
g * tan(56.0°) = (v^2) / r
6. Finally, solve for v by rearranging the equation:
v = sqrt(g * r * tan(56.0°))
7. Substitute the known values into the equation:
v = sqrt((9.8 m/s^2) * (12.0 m) * tan(56.0°))
8. Calculate the value of v using a calculator or mathematical software:
v ≈ 17.6 m/s
Therefore, the speed of the planes is approximately 17.6 m/s.