Question

Two wires are tied to the 1.5 kg sphere shown in the figure. (Figure 1) The sphere revolves in a horizontal circle at constant speed. For what speed is the tension the same in both wires? Express your answer to two significant figures and include the appropriate units. What is the tension? Express your answer to two significant figures and include the appropriate units.

Answer 1

To find the speed at which the tension is the same in both wires, we can equate the net force to the centripetal force and solve for the speed. The speed is given by v = sqrt((rg - g) / m). To find the tension, we can substitute the speed into the equation for the net force and solve for T. The tension is given by T = (mg - mrg + g^2) / (rg - g).

Step-by-step explanation:

To find the speed at which the tension is the same in both wires, we need to consider the forces acting on the sphere. Let's assume that the tension in one wire is T and the tension in the other wire is also T. The weight of the sphere is given by the equation W = mg, where m is the mass of the sphere and g is the acceleration due to gravity.

The net force acting on the sphere towards the center of the circle is given by F_net = T - W. Since the sphere is moving at a constant speed in a horizontal circle, the net force must be equal to the centripetal force, which is given by F_cen = (mv^2)/r, where m is the mass of the sphere, v is the speed of the sphere, and r is the radius of the circle.

Equating the net force and the centripetal force, we get T - mg = (mv^2)/r. Rearranging the equation, we can solve for v to find the speed at which the tension is the same in both wires:

v = sqrt((rg - g) / m)

Now, to find the tension, we can substitute the value of v into the equation for the net force:

T = (mv^2)/r + mg

Substituting the value of v from the previous equation, we can solve for T:

T = (mg - mrg + g^2) / (rg - g)

Answer 2

The speed at which the tension is the same in both wires is independent of the speed.

Step-by-step explanation:

To find the speed at which the tension is the same in both wires, we need to consider the forces acting on the sphere. The tension in each wire is equal to the centripetal force acting on the sphere: Tension = Centripetal Force. The centripetal force is given by the equation Fc = mv²/r, where m is the mass of the sphere, v is the velocity of the sphere, and r is the radius of the circular path. Since the mass and radius are given, we can rearrange the equation to solve for the velocity: v = √(Tension * r / m).

The tension in each wire can be found by setting the equations for the centripetal forces in the two wires equal to each other and solving for the tension: T₁ = T₂. Substituting the expressions for the tensions from the previous equation, we get √(Tension * r / m) = √(Tension * r / m). This means that the tension in both wires will be the same at any velocity. Therefore, the speed at which the tension is the same in both wires is independent of the speed.