As your bus rounds a flat curve at constant speed, a package with mass 0.300 , suspended from the luggage compartment of the bus by a string 46.0 long, is found to hang at rest relative to the bus, with the string making an angle of 30.0 with the vertical. In this position, the package is 50.0 from the center of curvature of the curve.

Question

As your bus rounds a flat curve at constant speed, a package with mass 0.300 , suspended from the luggage compartment of the bus by a string 46.0 long, is found to hang at rest relative to the bus, with the string making an angle of 30.0 with the vertical. In this position, the package is 50.0 from the center of curvature of the curve.

Answer 1

To determine the tension in the string, we need to consider the forces acting on the package in the rotating reference frame of the bus.

The first force acting on the package is the gravitational force, which points straight downward. Its magnitude is given by the mass of the package, multiplied by the acceleration due to gravity (9.8 m/s^2).

The second force acting on the package is the tension in the string, which points toward the center of the circular motion. This tension force is responsible for providing the centripetal force that keeps the package moving in a circular path.

Since the package hangs at rest relative to the bus, these two forces must be equal in magnitude but opposite in direction. Therefore, we can set up the following equation:

Tension = Gravitational force

To calculate the gravitational force, we use the equation:

Gravitational force = mass x acceleration due to gravity

Gravitational force = 0.300 kg x 9.8 m/s^2

Gravitational force = 2.94 N

Now, to find the tension in the string, we can use trigonometry. The vertical component of the tension is equal to the gravitational force (2.94 N), and the horizontal component of the tension can be found by using the sine function:

Horizontal component of tension = Tension x sin(angle)

Horizontal component of tension = Tension x sin(30.0°)

To isolate the tension, we rearrange the equation:

Tension = Horizontal component of tension / sin(angle)

Tension = (Tension x sin(30.0°)) / sin(30.0°)

Tension = Tension

Since the tension appears on both sides of the equation, we realize that the tension can have any value. This means that the package will hang at rest relative to the bus with any tension value, as long as the tension is equal to or greater than the gravitational force acting on the package. Hence, there is no unique answer for this particular question.