Your friend's 13.9g graduation tassel hangs on a string from his rearview mirror. When he accelerates from a stoplight, the tassel deflects backward toward the rear of the car at an angle of 5.47∘ relative to the vertical. Find the tension in the string holding the tassel. At what angle to the vertical will the tension in the string be twice the weight of the tassel?

Question

Your friend's 13.9g graduation tassel hangs on a string from his rearview mirror. When he accelerates from a stoplight, the tassel deflects backward toward the rear of the car at an angle of 5.47∘ relative to the vertical. Find the tension in the string holding the tassel. At what angle to the vertical will the tension in the string be twice the weight of the tassel?

Answer 1

To find the tension in the string holding the tassel, we can analyze the forces acting on it. The two main forces are the tension force in the string and the force due to gravity:

1. The weight force due to gravity is given by the equation: F_gravity = m * g, where m is the mass of the tassel, and g is the acceleration due to gravity (approximately 9.8 m/s^2).

2. The tension force in the string pulls the tassel upward and is responsible for the deflection angle. Let's denote it as T.

Using this information, we can break down the forces acting on the tassel into vertical and horizontal components:

- The vertical component of the tension force (T_y) must balance the weight force: T_y = m * g.

- The horizontal component of the tension force (T_x) causes the tassel to deflect backward. It is related to the vertical component by the tangent of the deflection angle: T_x = T * tan(angle).

Now we can find the tension in the string:

1. Calculate the vertical component of the tension force using T_y = m * g, where m is the mass of the tassel and g is the acceleration due to gravity (approximately 9.8 m/s^2).

2. Calculate the horizontal component of the tension force using T_x = T * tan(angle), where T is the tension force and angle is the deflection angle in radians (convert degrees to radians by dividing by 180 and multiplying by π).

3. Use the Pythagorean theorem to find the magnitude of the tension force T: T^2 = T_x^2 + T_y^2.

4. Once you find the tension force T, compare it to twice the weight of the tassel (2 * m * g). If T is twice the weight of the tassel, then the tension in the string is twice the weight.

To summarize:
1. T_y = m * g
2. T_x = T * tan(angle)
3. T^2 = T_x^2 + T_y^2
4. Compare T to 2 * m * g to find the angle to the vertical.

By following these steps, you should be able to determine the tension in the string holding the tassel and find the angle at which the tension is twice the weight of the tassel.