A gymnast is swinging on a high bar. The distance between his waist and the bar is 1.26 m, as the drawing shows. At the top of the swing his speed is momentarily zero. Ignoring friction and treating the gymnast as if all of his mass is located at his waist, find his speed at the bottom of the swing.
To find the gymnast's speed at the bottom of the swing, we can use the principle of conservation of mechanical energy. At the top of the swing, the gymnast's gravitational potential energy is at its maximum, while his kinetic energy is at its minimum (since his speed is momentarily zero).
1. First, let's determine the height of the swing. As the distance between his waist and the bar is given as 1.26 m, this indicates the radius of the swing. Since the swing is in the shape of a circular arc, the height of the swing can be determined using the formula for the length of an arc:
length of arc = radius * angle in radians.
2. Assuming the swing makes a complete cycle (360 degrees or 2π radians), the length of the arc is equal to the circumference of the circle. So, we can calculate the height using the formula:
height = 2 * radius * π.
3. Once we have the height of the swing, we can calculate the gravitational potential energy at the top of the swing using the formula:
gravitational potential energy = mass * gravitational acceleration * height.
4. At the top of the swing, the gravitational potential energy is converted entirely into kinetic energy at the bottom of the swing. So, the kinetic energy at the bottom of the swing is equal to the gravitational potential energy at the top.
5. The formula for kinetic energy is:
kinetic energy = 0.5 * mass * velocity^2.
6. Rearranging the equation to solve for velocity:
velocity = sqrt(2 * gravitational potential energy / mass).
Using this approach, you can now find the gymnast's speed at the bottom of the swing using the given information about the distance between his waist and the bar.