A train traveling at a constant speed rounds a curve of radius 250 m. A chandelier suspended from the ceiling swings out to an angle of 15.5° throughout the turn. What is the speed of the train?
what equation should i use?
You need to think it through and solve two equations. The fixture or chain that holds the chandelier exerts a tension force T upon it that is inclined 15.5 degrees to the vertical. The vertical component supports the weight, M g, and the horizontal component provides the centripetal force, M V^2/R . Thus
T cos15.5 = M g, and
T sin15.5 = M V^2/R
You can divide the last equation by the first to eliminate T and M
tan15.5 = V^2/(gR) = 0.2173
Solve for V, in m/s
To determine the speed of the train, you can use the centripetal acceleration equation. The centripetal acceleration is given by the formula:
a = (v^2) / r
Where:
a = Centripetal acceleration
v = Velocity (speed) of the train
r = Radius of the curve (250 m)
Since the problem states that the chandelier swings out to an angle of 15.5° throughout the turn, we can relate that angle to the velocity of the train. When an object swings through an angle, it experiences a centripetal force that keeps it moving in a circular path.
The angle can be related to the centripetal acceleration using the equation:
a = g * tan(theta)
Where:
g = Acceleration due to gravity (approximately 9.8 m/s²)
theta (θ) = Angle (15.5° in this case)
By equating these two equations and substituting the given values, we can find the velocity of the train. So, the equation you should use is:
(v^2) / r = g * tan(theta)
Rearranging this equation, you can solve for v:
v = sqrt(r * g * tan(theta))
Plug in the given values and calculate the answer.