Question

Two masses hang from two strings of equal length that are attached to the ceiling of a car. One mass is over the driver's seat; the other is over the passenger's seat. As the car makes a sharp turn, both masses swing away from the center of the turn. In their resulting positions, will they be farther apart, closer together, or the same distance apart as they were when the car wasn't turning?

Short answer

Answer: The masses become farther apart after the car makes a sharp turn.

Step-by-step solution

Identify the forces acting on the masses

Before the car turns, we need to identify the forces acting on each mass. When the car is not turning, the only forces acting on each mass are the tension in the string due to the weight of the mass, and the gravitational force. As the car makes a sharp turn, there is an additional force acting on the masses due to the centripetal acceleration, which is directed toward the center of the turn.

Analyze the equilibrium position of the masses after the car turns

When the car makes the sharp turn, the masses will tend to swing away from the center of the turn due to the centripetal force. As a result, the equilibrium position of each mass will change. Let's denote the angles between the strings and the vertical direction before and after the turn as θ₁ and θ₂ respectively. We don't have actual values for these angles, but we know that θ₂ is greater than θ₁, as the mass swings away from the center of the turn.

Compare the distance between the masses before and after the turn

To compare the initial distance from when the car is not turning (D₁) and the distance after the car makes a sharp turn (D₂), we will use the law of cosines. The law of cosines relates the length of the sides of a triangle to the cosine of one of its angles. The distances D₁ and D₂ can be found using the following equations: D₁ = 2 * L * sin(θ₁ / 2) D₂ = 2 * L * sin(θ₂ / 2), where L is the length of the strings. Since θ₂ > θ₁, we have sin(θ₂ / 2) > sin(θ₁ / 2). Then, multiplying both sides by 2 * L, we get: D₂ > D₁.

Conclusion

Based on our comparison of the initial and final distances, we can conclude that the masses will be farther apart after the car makes a sharp turn than they were when the car wasn't turning.

Key concepts

Equilibrium Position

In the context of physics and the problem at hand, an equilibrium position refers to the state where a mass remains at rest or moves with a constant velocity because all the acting forces balance each other out. In scenarios involving strings and pendulums, such as the masses hanging inside a car, equilibrium is achieved when the net force on the masses is zero. This means that, when the car is stationary or moving at a constant velocity, the gravitational force pulling the masses down is perfectly counteracted by the tension in the strings pulling them up.

However, when the car makes a turn, things change. The masses experience a centripetal force that pulls them toward the center of the turn, altering their equilibrium position. This centripetal force is not counteracted by any other force, which results in the masses swinging away from the turn's center until a new equilibrium position is reached. Here, the tension in the strings provides the necessary centripetal force to keep the masses in a circular path, making the swinging outward a natural response to maintain equilibrium in the context of the circular motion.

Law of Cosines

The law of cosines is a fundamental principle in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. This law is particularly useful when you are dealing with non-right triangles, where the more familiar Pythagorean theorem does not apply. It is often used to solve for unknown lengths or angles within a given triangle.

Mathematically, the law of cosines is expressed as: \[\begin{equation}c^2 = a^2 + b^2 - 2ab \cdot \cos(\gamma)\end{equation}\] where \(c\) is the length of the side opposite the angle \(\gamma\), and \(a\) and \(b\) are the lengths of the other two sides. In the problem provided, the law of cosines can be adapted to compare the distances between two swinging masses within the car before and after the turn. The change in angles as the car turns changes the shape of the triangle formed by the strings and the horizontal line connecting the masses, thus altering the distance between them as calculated by this law.

Centripetal Acceleration

Centripetal acceleration is a key concept in the realm of circular motion. It is the acceleration experienced by an object moving along a curved path that is directed towards the center of the curvature of the path. This is why, in circular motion, even if an object is moving at a constant speed, it still experiences acceleration because its direction is continuously changing.

The formula for centripetal acceleration (\(a_c\)) is: \[\begin{equation}a_c = \frac{v^2}{r}\end{equation}\]where \(v\) is the object's speed and \(r\) is the radius of the circular path. In our given problem, when the car is turning sharply, the hanging masses experience this inward acceleration, which changes their equilibrium position and causes the masses to swing outward. The direction of the centripetal acceleration is critical here-it is always directed towards the center of the turn, which for the masses in the car, is the direction opposite to the swing caused by the turn.