A stone is tied to a string (length = 0.700 m) and whirled in a circle at the same constant speed in two different ways. First, the circle is horizontal and the string is nearly parallel to the ground. Next, the circle is vertical. In the vertical case the maximum tension in the string is 16.0% larger than the tension that exists when the circle is horizontal. Determine the speed of the stone.

Horizontal

F = m v^2/r

Vertical
Fmax = m g + mv^2/r

m g + m v^2/r = 1.16 m v^2/r

g = .16 v^2/r

.16 v^2 = 9.8(.7)
solve

Why did the stone get dizzy? Because it was going round and round in circles! Now, let's solve this riddle.

Let's assume the tension in the string when the circle is horizontal is T (in Newtons). As given in the problem, the tension in the string when the circle is vertical is 16.0% larger than the tension in the horizontal case.

So, the tension in the vertical case is 1.16T (in Newtons).

Now, let's consider the horizontal case. In this case, the only force acting on the stone is the tension in the string. And since the stone is moving in a circle, there must be a centripetal force acting on it.

According to Newton's second law of motion, the centripetal force is given by the equation:

F = m * a

Where F is the centripetal force, m is the mass of the stone, and a is the centripetal acceleration.

The centripetal force is provided by the tension in the string, so we can write:

T = m * a

Now, let's consider the vertical case. In this case, there are two forces acting on the stone: the tension in the string (upwards) and the force of gravity (downwards).

Since the stone is moving in a vertical circle, there must be a net force acting towards the center of the circle. This net force is responsible for providing the centripetal force.

So, we can write:

T + mg = m * a

Where T is the tension in the string, m is the mass of the stone, g is the acceleration due to gravity, and a is the centripetal acceleration.

Now, let's find a relationship between a in the horizontal case and a in the vertical case.

In the horizontal case, the stone is moving along a circular path parallel to the ground. So, the centripetal acceleration is given by:

a = v^2 / r_h

Where v is the speed of the stone and r_h is the radius of the horizontal circle.

Similarly, in the vertical case, the centripetal acceleration is given by:

a = v^2 / r_v

Where r_v is the radius of the vertical circle.

Since r_v > r_h (as the string length is the hypotenuse), we can write:

v^2 / r_v > v^2 / r_h

This implies that a_v > a_h (as both sides are divided by m).

Now, we can write the equations for tension in terms of a_h and a_v:

T = m * a_h

1.16T = m * a_v

Dividing these two equations, we get:

1.16T / T = a_v / a_h

1.16 = a_v / a_h

Since a_v is greater than a_h, we can write:

1.16 > 1

This is a contradiction. Therefore, there is no possible solution to this problem.

Looks like the stone is playing mind games with us by creating an unsolvable riddle!

To determine the speed of the stone, we need to utilize the concept of centripetal force.

In the horizontal case, when the stone is whirled in a circle parallel to the ground, the tension in the string is providing the necessary centripetal force to keep the stone moving in a circular path. Let's denote this tension as T_horizontal.

In the vertical case, when the stone is whirled in a vertical circle, the tension in the string is providing both the centripetal force and the force of gravity acting on the stone. Since the maximum tension in the string is 16.0% larger than the tension in the horizontal case, we can denote the tension in the vertical case as T_vertical = 1.16 * T_horizontal.

Now, let's analyze the forces acting on the stone in the vertical case. We have two forces:

1. T_vertical: The tension in the string acting upwards.
2. mg: The gravitational force acting downwards, where m is the mass of the stone and g is the acceleration due to gravity.

The net force in the vertical direction is given by:

T_vertical - mg = m * (v_vertical)^2 / r

where v_vertical is the speed of the stone in the vertical case and r is the radius of the circle (length of the string).

Since the speed of the stone is the same in both cases, we can equate the centripetal force expressions for the horizontal and vertical cases:

T_horizontal = m * (v_horizontal)^2 / r
T_vertical = m * (v_vertical)^2 / r

Substituting the value of T_vertical, we get:

1.16 * T_horizontal = m * (v_vertical)^2 / r

Now, divide the above equation by the first equation:

(1.16 * T_horizontal) / T_horizontal = (m * (v_vertical)^2 / r) / (m * (v_horizontal)^2 / r)

1.16 = (v_vertical)^2 / (v_horizontal)^2

Take the square root of both sides:

sqrt(1.16) = v_vertical / v_horizontal

Solving for v_vertical / v_horizontal:

v_vertical / v_horizontal = sqrt(1.16)

To find the speed of the stone, we need to know the ratio of v_vertical to v_horizontal. If we assume that the lengths of the string in both cases are the same (0.700 m), then v_vertical and v_horizontal are related by the angle at which the string makes with the ground.

If we let θ be the angle between the string and the ground, then:

sin(θ) = v_vertical / v_horizontal

Since the angle in the horizontal case is nearly parallel to the ground, we have:

sin(θ) = 0.999 ≈ 1

Now, we can substitute this value into the equation:

1 = sqrt(1.16)

Squaring both sides:

1 = 1.16

This is a contradiction, so our assumption that the lengths of the string in both cases are the same is incorrect. Therefore, we cannot determine the speed of the stone with the given information.

I do now