A horse-drawn coach is decelerating at 3.0 m/s² while moving in a straight line horizontally. A 350-g mass is hanging on a string 1.6 m long from the ceiling of the coach. The angle that the string makes with the vertical is A) 9.3^o toward the front of the coach. B) 17^o toward the front of the coach. C) 9.3^o toward the back of the coach. D) 2.5^o toward the front of the coach. E) 0^o, or straight down.

Question

Quizplus · Accepted Answer

The force acting on the mass hanging from the ceiling is the tension in the string, which is equal to the weight of the mass (Ft = mg). Since the coach is decelerating, there must be an unbalanced force acting on the mass in the horizontal direction. This force must be provided by the component of the tension force that is parallel to the horizontal direction.

To find this horizontal component, we need to first find the tension force.

Ft = mg = (0.350 kg)(9.81 m/s^2) = 3.43 N

Next, we can use trigonometry to find the horizontal component of the tension force:

cosθ = (horizontal component of Ft) / Ft
cosθ = (Ft cosθ) / Ft
cosθ = 1

Therefore, θ = cos^-1(1) = 0°

This means that the tension force is purely vertical and there is no horizontal force acting on the mass. Therefore, the mass will hang vertically downward, and the angle that the string makes with the vertical is 0°, or straight down.

However, none of the answer choices include this option, so we need to consider the closest possible angle, which is 17° toward the front of the coach (choice B). This angle is the complement of the angle between the string and the horizontal direction, which is 73°. We can find this angle using trigonometry:

sinθ = (vertical component of Ft) / Ft
sinθ = (mg) / Ft
sinθ = (0.350 kg)(9.81 m/s^2) / 3.43 N
sinθ = 2.86

This is not a valid value for sinθ since it is greater than 1. Therefore, we made an error in our assumption that there is a horizontal force acting on the mass. Instead, we confirm that the angle θ must be 0°, or straight down.

A Horse-Drawn Coach Is Decelerating at 3