Kelli weighs 392 N, and she is sitting on on a playground swing that hangs 0.4 m above the ground. Her mom pulls the swing back and releases it when the seat is 1.1 m above the ground. If Kelli moves through the lowest point at 1.5 m/s, how much work was done on the swing by friction?
To find the work done on the swing by friction, we need to calculate the change in mechanical energy.
The mechanical energy of an object consists of kinetic energy (KE) and potential energy (PE). In this case, the swing starts at an initial height and ends at a final height, with an initial velocity.
1. Calculate the initial gravitational potential energy:
PE_initial = m * g * h_initial
PE_initial = 392 N × 0.4 m
PE_initial = 156.8 J
2. Calculate the final gravitational potential energy:
PE_final = m * g * h_final
PE_final = 392 N × 1.1 m
PE_final = 431.2 J
3. Calculate the initial kinetic energy:
KE_initial = 0.5 * m * v_initial^2
KE_initial = 0.5 * 392 N * (1.5 m/s)^2
KE_initial = 441 J
4. Calculate the final kinetic energy:
KE_final = 0.5 * m * v_final^2
KE_final = 0.5 * 392 N * 0^2
KE_final = 0 J (since the swing is at its lowest point, the velocity is zero)
5. Calculate the change in mechanical energy:
ΔE = (PE_final + KE_final) - (PE_initial + KE_initial)
ΔE = (431.2 J + 0 J) - (156.8 J + 441 J)
ΔE = 431.2 J - 598.8 J
ΔE = -167.6 J
The negative sign indicates a loss of energy, implying the work done on the swing is by friction.
Therefore, the work done on the swing by friction is 167.6 Joules.
To calculate the work done on the swing by friction, we need to determine the change in mechanical energy from the highest point to the lowest point. The change in mechanical energy is equal to the work done by all forces involved, including friction.
The mechanical energy at the highest point consists of the potential energy and the kinetic energy. The potential energy is given by the equation:
PE = mgh
Where:
PE = Potential energy
m = Mass (which we can calculate using the weight of Kelli)
g = Acceleration due to gravity (approximated as 9.8 m/s^2)
h = Height above the reference point (in this case, the ground)
The kinetic energy is given by the equation:
KE = 1/2 * mv^2
Where:
KE = Kinetic energy
m = Mass
v = Velocity
In this case, we are given the height at the highest point (1.1 m) and the velocity at the lowest point (1.5 m/s). We need to calculate the mass of Kelli using her weight (392 N).
Since weight (W) is equal to mass (m) multiplied by acceleration due to gravity (g), we can rearrange the equation to solve for mass:
m = W / g
Now, let's plug in the given values:
m = 392 N / 9.8 m/s^2
m = 40 kg
Now, we can calculate the potential energy (PE) at the highest point:
PE = mgh
PE = 40 kg * 9.8 m/s^2 * 1.1 m
PE = 431.2 J
Next, we calculate the kinetic energy (KE) at the lowest point:
KE = 1/2 * mv^2
KE = 1/2 * 40 kg * (1.5 m/s)^2
KE = 45 J
The work done by friction is equal to the change in mechanical energy:
Work by friction = PE - KE
Work by friction = 431.2 J - 45 J
Work by friction = 386.2 J
Therefore, the work done on the swing by friction is 386.2 Joules.