A train traveling at a constant speed rounds a curve of radius 200 m. A chandelier suspended from the ceiling swings out to an angle of 17.0° throughout the turn. What is the speed of the train?
Again, use your formulas. Simple plug-n-chug with radial/angular force.
To determine the speed of the train, we can use the concept of centripetal force. The centripetal force acting on the chandelier is responsible for keeping it in circular motion as the train goes around the curve.
The centripetal force (Fc) acting on an object moving in a circular path is given by the equation:
Fc = (m * v^2) / r
Where:
- Fc is the centripetal force
- m is the mass of the object
- v is the velocity (speed) of the object
- r is the radius of the circular path
In this case, the chandelier is acting as the object and the force is provided by gravity exerted on it at an angle of 17.0°. Therefore, the formula can be modified to:
Fc = (m * g * sinθ) = (m * v^2) / r
Where:
- m is the mass of the chandelier
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- θ is the angle swung out by the chandelier
- r is the radius of the curve
We want to solve for v, the speed of the train.
Rearranging the formula, we get:
v^2 = (r * g * sinθ)
The radius of the curve is given as 200 m and the angle θ is given as 17°.
Plugging these values into the formula:
v^2 = (200 m) * (9.8 m/s^2) * sin(17°)
Now, calculate the value of sin(17°) and the product of the other terms:
v^2 = (200 m) * (9.8 m/s^2) * 0.2924
Next, multiply the terms on the right side of the equation:
v^2 ≈ 5681.6 m^2/s^2
To find v, take the square root of both sides:
v ≈ √(5681.6 m^2/s^2)
Calculating this expression, we find:
v ≈ 75.405 m/s
So, the speed of the train is approximately 75.405 m/s.