A passenger on a train notices that a 2.00 kg mass is displaced 10.0° from the vertical as the train
accelerates. (a) What is the acceleration of the train? (b) What would the acceleration be if the
mass was 5.00 kg?
c. What would angle have to be to indicate a horizontal acceleration equal to the
acceleration of gravity?
d. Could the acceleration ever be great enough to have the angle be 90.0° -- that is,
have the cord suspending the mass be horizontal? Explain.
To answer these questions, we need to consider the forces acting on the mass and the relationship between these forces and the acceleration of the train.
Let's start with the first question: (a) What is the acceleration of the train?
Given:
Mass of the object, m = 2.00 kg
Displacement angle, θ = 10.0°
When the object is displaced at an angle, there are two forces acting on it: gravitational force (mg) and the tension force in the string (T).
The gravitational force can be calculated using the formula:
Force due to gravity, Fg = mg
The tension force T can be determined using trigonometry. The vertical component of the tension force can balance the gravitational force:
Vertical component of T = T * sin(θ)
This vertical component should balance the gravitational force, so we have:
T * sin(θ) = mg
Now, if we accelerate the train, there will be an additional tension force created due to the acceleration. Assuming the train accelerates horizontally, the horizontal component of the tension force will provide the necessary force:
Horizontal component of T = T * cos(θ)
This horizontal component should be equal to the mass times the acceleration of the train:
T * cos(θ) = ma
Solving these two equations simultaneously, we can find the acceleration of the train (a).
To answer the second question: (b) What would the acceleration be if the mass was 5.00 kg?
We can use the same equations as above, just replacing the mass (m) with the new value (5.00 kg).
To answer the third question: (c) What would the angle have to be to indicate a horizontal acceleration equal to the acceleration of gravity?
We need to find the angle at which the horizontal component of the tension force is equal to the gravitational force. This means we need to solve the equation: T * cos(θ) = mg, where a = g (acceleration of gravity).
To answer the fourth question: (d) Could the acceleration ever be great enough to have the angle be 90.0° (the cord suspending the mass be horizontal)? Explain.
In order for the angle to be 90.0°, the tension force (T) would need to be zero because the cosine of 90.0° is zero. This would mean that the horizontal component of the tension force (T * cos(90.0°)) cannot provide any force to balance the gravitational force (mg). Therefore, the acceleration of the train cannot be high enough for the angle to be 90.0°.
To summarize, to answer these questions, you will need to use the equations for forces, trigonometry, and the relationship between forces and acceleration.