Question

Consider a game in which N children position themselves at equal distances around the circumference of a circle. At the center of the circle is a rubber tire. Each child holds a rope attached to the tire and, at a signal, pulls on his on her rope. All children exert forces of the same magnitude F . In the case N=2, it is easy to see that the net force on the tire will be zero because the two oppositely directed force vectors add to zero. Similarly, if N=4,6, or any even integer, the resultant force on the tire must be zero because the forces exerted by each pair of oppositely positioned children will cancel. When an odd number of children are around the circle, it is not as obvious whether the total force on the central tire will be zero. Calculate the net force on the tire in the case N=3 by adding the components of the three force vectors. Choose the x axis to lie along one of the ropes.

Answer 1

In the case where N = 3, we have three children positioned at equal distances around the circumference of the circle. Each child exerts a force of the same magnitude F on the tire, and we need to calculate the net force on the tire.

To calculate the net force, we can break down each force vector into its x and y components. Since the x-axis is chosen to lie along one of the ropes, we can consider the forces in the x-direction only. The forces in the y-direction will cancel out.

Let's denote the angle between each force vector and the positive x-axis as θ. Since the children are equally spaced, each angle θ will be 120 degrees (360 degrees divided by 3 children).

Now, let's break down each force vector into its x and y components. The x-component of each force can be calculated using cosine, while the y-component can be calculated using sine.

1. Force 1:

- x-component: F * cos(θ)

- y-component: F * sin(θ)

2. Force 2:

- x-component: F * cos(θ)

- y-component: -F * sin(θ) (negative sign because the force is in the opposite direction)

3. Force 3:

- x-component: -2 * F * cos(θ) (negative sign because the force is in the opposite direction)

- y-component: 0 (no y-component since the force is along the x-axis)

To calculate the net force in the x-direction, we add up the x-components of all the forces:

Net force in the x-direction = (F * cos(θ)) + (F * cos(θ)) + (-2 * F * cos(θ))

= 2 * F * cos(θ) - 2 * F * cos(θ)

= 0

Since the net force in the x-direction is zero, we can conclude that the total force on the central tire when N = 3 is zero.

Therefore, the net force on the tire in the case N = 3 is zero.