A 40 kg child sits in a swing suspended with 2.5 m long ropes. The swing is held aside so that the ropes make an angle of 15 degrees with the vertical. Use the conservation of mechanical energy to determine the speed the child will have at the bottom of the arc when she is let go.
To determine the speed the child will have at the bottom of the arc when she is let go, we can use the conservation of mechanical energy. The mechanical energy of a system remains constant as long as no external forces, such as friction or air resistance, are acting on it.
The mechanical energy of the system consists of two components: the gravitational potential energy and the kinetic energy. At the top of the swing arc, when the child is held aside, all of the gravitational potential energy is converted to kinetic energy as the child swings downward. At the bottom of the arc, all of the kinetic energy is converted back to gravitational potential energy.
To apply the conservation of mechanical energy, we can equate the initial gravitational potential energy to the final kinetic energy.
1. Calculate the initial height:
The initial height is the vertical distance from the highest point to the bottom of the swing arc. We can determine this height using trigonometry. Given that the ropes make an angle of 15 degrees with the vertical and each rope is 2.5 m long, the vertical component of the rope can be calculated as follows:
Vertical component = 2.5 m * sin(15 degrees)
2. Calculate the initial gravitational potential energy:
The initial gravitational potential energy is given by the equation:
Gravitational potential energy = mass * gravity * height
The mass of the child is given as 40 kg, and the acceleration due to gravity is approximately 9.8 m/s^2. Substituting the calculated height, we get:
Initial gravitational potential energy = 40 kg * 9.8 m/s^2 * (vertical component)
3. Calculate the final kinetic energy:
The final kinetic energy can be calculated using the equation:
Kinetic energy = 1/2 * mass * velocity^2
At the bottom of the swing arc, the child's potential energy is converted completely to kinetic energy. Therefore, the final kinetic energy is equal to the initial gravitational potential energy.
4. Solve for velocity:
Equating the initial gravitational potential energy to the final kinetic energy, we have:
Initial gravitational potential energy = 1/2 * mass * velocity^2
Substituting the values, we can solve for velocity:
40 kg * 9.8 m/s^2 * (vertical component) = 1/2 * 40 kg * velocity^2
Simplifying, we find:
(vertical component) = 1/2 * velocity^2
To isolate velocity, we can rearrange the equation:
velocity^2 = 2 * (vertical component)
Taking the square root of both sides gives us:
velocity = √(2 * (vertical component))
Now, substitute the calculated value of (vertical component) and solve for the velocity.
Note: Make sure to use the appropriate units throughout the calculations.