A 80- kg child sits in a swing suspended with 2.6- m-long ropes. The swing is held aside so that the ropes make an angle of 46 o with the vertical. Use conservation of energy to determine the speed the child will have at the bottom of the arc when she is let go.
To determine the speed of the child at the bottom of the arc when she is let go, we can use the principle of conservation of mechanical energy. The mechanical energy of the system remains constant, assuming no energy losses to factors such as friction or air resistance.
Here's how we can solve it step by step:
Step 1: Determine the gravitational potential energy at the starting position.
Gravitational potential energy (PE) is given by the formula: PE = m * g * h, where m is the mass of the child, g is the acceleration due to gravity, and h is the height.
In this case, the height is the vertical distance from the starting position to the lowest point of the swing's arc.
The height can be calculated using trigonometry: h = length of the ropes * sin(angle).
Given:
Mass of the child (m) = 80 kg
Length of the ropes (l) = 2.6 m
Angle (θ) = 46 degrees
h = l * sin(θ) = 2.6 * sin(46) = 1.86 m
Step 2: Determine the kinetic energy at the bottom of the arc.
At the lowest point of the swing's arc, all the potential energy is converted into kinetic energy.
The kinetic energy (KE) is given by the formula: KE = (1/2) * m * v^2, where m is the mass of the child and v is the velocity.
Since energy is conserved, the gravitational potential energy at the starting position is equal to the kinetic energy at the bottom of the arc.
PE = KE
m * g * h = (1/2) * m * v^2
Step 3: Solve for velocity (v).
Rearranging the equation from the previous step to solve for velocity (v) gives:
v^2 = 2 * g * h
v = sqrt(2 * g * h)
Substituting the known values into the equation gives:
v = sqrt(2 * 9.8 * 1.86)
v ≈ 6.50 m/s
Therefore, when the child is let go at the bottom of the arc, she will have a speed of approximately 6.50 m/s.