A 35kg child is swinging on a swing with length 2.8m to the bottom of the seat. At the highest point of the swing, the rope makes an angle of 42 degrees from vertical. Calculate the speed of the child at the bottom of the swing. Assume no thermal energy is created. Calculate the force the child exerts on the seat at the bottom of the swing.
at 42 degrees, the change in height of the swing = 2.8*(1-cos42)
speed at bottom=above height*mass*g
forceatbottom=masschild*V^2/2.8 + masschild*g
To calculate the speed of the child at the bottom of the swing, we can use the conservation of energy. At the highest point, gravitational potential energy is converted into kinetic energy at the bottom.
Step 1: Calculate the potential energy at the highest point:
Potential Energy (PE) = mass * gravity * height
PE = 35kg * 9.8m/s^2 * 2.8m = 960.4 J
Step 2: Calculate the kinetic energy at the bottom:
Kinetic Energy (KE) = 1/2 * mass * velocity^2
Step 3: equate PE to KE:
960.4 J = 1/2 * 35kg * velocity^2
1920.8 J = 35kg * velocity^2
Step 4: Solve for velocity:
velocity^2 = 1920.8 J / 35kg
velocity^2 ≈ 54.88 m^2/s^2
velocity ≈ √(54.88) ≈ 7.41 m/s
Therefore, the speed of the child at the bottom of the swing is approximately 7.41 m/s.
To calculate the force the child exerts on the seat at the bottom of the swing, we can use Newton's second law of motion, which states that force equals mass times acceleration.
Step 1: Calculate the acceleration:
Since the child is moving in a circle, the acceleration is given by:
acceleration = velocity^2 / radius
Step 2: Determine the radius:
The radius can be found using the length of the swing and the angle made by the rope with the vertical.
radius = length * sin(angle)
radius = 2.8m * sin(42°)
radius ≈ 2.8m * 0.6691 ≈ 1.87m
Step 3: Calculate acceleration:
acceleration = (7.41 m/s)^2 / 1.87m
acceleration ≈ 29.27 m/s^2
Step 4: Calculate the force:
force = mass * acceleration
force = 35kg * 29.27 m/s^2
force ≈ 1024.45 N
Therefore, the force the child exerts on the seat at the bottom of the swing is approximately 1024.45 N.
To calculate the speed of the child at the bottom of the swing, we can use the conservation of mechanical energy, assuming no thermal energy is created.
The mechanical energy of the child on the swing consists of two components: potential energy (PE) and kinetic energy (KE).
1. First, let's calculate the potential energy at the highest point of the swing (when the rope makes a 42-degree angle from vertical).
The potential energy (PE) is given by the equation: PE = m * g * h, where m is the mass of the child (35kg), g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height of the swing seat above the lowest point of the swing (2.8m).
PE = 35 kg * 9.8 m/s² * 2.8m = 960.4 J (Joules)
2. Next, we can calculate the kinetic energy (KE) at the bottom of the swing (when the child is at the lowest point of the swing).
The kinetic energy (KE) is given by the equation: KE = (1/2) * m * v², where m is the mass of the child and v is the velocity of the child at the bottom.
Since the mechanical energy is conserved, the potential energy at the highest point must equal the kinetic energy at the lowest point. Thus, we can equate the two equations:
PE = KE
960.4 J = (1/2) * 35 kg * v²
Solving for v, we have:
v² = 2 * (960.4 J) / (35 kg)
v² = 54.8686 m²/s²
Therefore, v ≈ 7.41 m/s (meters per second) at the bottom of the swing.
Now let's calculate the force the child exerts on the seat at the bottom of the swing.
At the bottom of the swing, the net force acting on the child is the centripetal force (Fc) directed towards the center of the circular motion.
The centripetal force is given by the equation: Fc = m * v² / r, where m is the mass of the child, v is the velocity, and r is the radius of the swing (half the length of the swing seat).
In this case, the radius (r) is given as 2.8m / 2 = 1.4m.
Substituting the values into the equation:
Fc = 35 kg * (7.41 m/s)² / 1.4m
Fc = 35 kg * 54.8686 m²/s² / 1.4m
Fc ≈ 1373.21 N (Newtons)
Therefore, the child exerts a force of approximately 1373.21 N on the seat at the bottom of the swing.