A child is on a swing whose 4.0 m long chains make a maximum angle of 30 deg with the vertical. What is the child's maximum speed?

I've found problems relating to this but include mass or delta(x) in how far from stationary. This problem doesnt include mass. Help?

mgh = 1/2 mv^2

m cancels.
h = 4 - 4cos30 (draw the triangle, you'll see)

To find the maximum speed of the child on the swing, we can use the concept of conservation of mechanical energy.

The key idea is that at the highest point of the swing, where the chains make a maximum angle of 30 degrees with the vertical, all the mechanical energy is in the form of gravitational potential energy. At this point, the kinetic energy is zero.

The gravitational potential energy of an object at height h is given by the equation:

PE = m * g * h

Where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s^2 on Earth), and h is the height above the reference point.

In this case, we are not given the mass of the child. However, since we are only interested in the maximum speed, we can assume that the child has negligible mass.

As a result, we can simplify the equation to:

PE = h * g

Next, since the maximum angle is 30 degrees, we can use trigonometry to find the height h. In a right-angled triangle, the vertical side is equal to the hypotenuse (the length of the chains) multiplied by the sine of the angle:

h = L * sin(theta)

Where L is the length of the chains (4.0 m in this case) and theta is the angle (30 degrees in this case).

Plugging in the values:

h = 4.0 * sin(30) = 2.0 m

Now we can calculate the gravitational potential energy at the highest point:

PE = 2.0 * 9.8 = 19.6 J

At the highest point, all the mechanical energy is in the form of potential energy. As the child swings downward, the potential energy is gradually converted into kinetic energy. At the lowest point, all the energy is in the form of kinetic energy, and the potential energy is zero.

The kinetic energy of an object is given by the equation:

KE = (1/2) * m * v^2

Where v is the velocity of the object.

Since the potential energy at the highest point is equal to the kinetic energy at the lowest point, we have:

PE = KE

19.6 J = (1/2) * m * v^2

Since we are assuming the child's mass is negligible, we can further simplify the equation to:

19.6 J = (1/2) * v^2

Now we can solve for v:

v^2 = (2 * 19.6) / 1

v^2 = 39.2

v = √39.2

v ≈ 6.26 m/s

Therefore, the child's maximum speed on the swing is approximately 6.26 m/s.