Cheerleaders at sporting events will often use a bullhorn to make themselves louder. A bullhorn is simply a hollowed out cone - you speak into the narrow end and your voice comes out the wide end, channeled by the cone so it is louder than it would be otherwise. At a sporting event, a cheerleader holds a bullhorn up to their mouth as in the figure below. The very tip of the bullhorn rests on their mouth and they use their hand on the other end to hold the bullhorn horizontal and still. We model the bullhorn as the rigid 2-d hollow cone of horizontal length 0.75 m, opening angle 30∘, and (surface) mass density of 0.4 kg/m2 (see figure). How much force in Newtons does the cheerleader exert at point A?

Question

Cheerleaders at sporting events will often use a bullhorn to make themselves louder. A bullhorn is simply a hollowed out cone - you speak into the narrow end and your voice comes out the wide end, channeled by the cone so it is louder than it would be otherwise. At a sporting event, a cheerleader holds a bullhorn up to their mouth as in the figure below. The very tip of the bullhorn rests on their mouth and they use their hand on the other end to hold the bullhorn horizontal and still. We model the bullhorn as the rigid 2-d hollow cone of horizontal length 0.75 m, opening angle 30∘, and (surface) mass density of 0.4 kg/m2 (see figure). How much force in Newtons does the cheerleader exert at point A?

Answer 1

To determine the force exerted by the cheerleader at point A, we need to consider the balance of forces acting on the bullhorn.

First, let's consider the weight of the bullhorn. The weight can be calculated using the mass density and the surface area of the bullhorn.

The mass density is given as 0.4 kg/m^2, and the bullhorn is a hollow cone with a certain length and opening angle. The surface area of the bullhorn can be calculated using the formula for the lateral surface area of a cone (excluding the base).

Let's start with finding the surface area of the bullhorn:

Surface Area = π * r * s

Here, r is the radius of the wide end of the cone, and s is the slant height. Since the bullhorn is described as a hollow cone, we can use similar triangles to find the slant height:

Tan(30°) = s / (0.75 m)

Solving for s, we get:

s = 0.75 m * Tan(30°)

Now we can calculate the radius of the wide end of the cone:

r = (0.75 m * Tan(30°)) * Tan(30°)

Using this radius, we can calculate the surface area:

Surface Area = π * r * (0.75 m * Tan(30°)) * Tan(30°)

Next, we can calculate the weight of the bullhorn:

Weight = Surface Area * mass density

Now, let's consider the forces acting on the bullhorn:

1. Weight: The weight of the bullhorn acts vertically downward at its center of mass.

2. Force exerted by the cheerleader at point A: This force needs to balance the weight and maintain the bullhorn in equilibrium.

Since the bullhorn is held horizontally, the weight is acting vertically downward. Therefore, the force exerted by the cheerleader at point A needs to counteract this vertical component of the weight.

To find the force exerted by the cheerleader at point A, we need to consider the vertical component of the weight. We can do this using trigonometry:

Vertical Component of Weight = Weight * Sin(30°)

Finally, the force exerted by the cheerleader at point A is equal in magnitude but opposite in direction to the vertical component of the weight. Therefore, the force exerted by the cheerleader at point A is:

Force at Point A = -Vertical Component of Weight

You can substitute the values into the above equations to find the numerical value for the force exerted by the cheerleader at point A.