02.08.2022 - 08:27

# A person can generate about 300 W of power on a treadmill. If the treadmill is inclined at 2% and a 70-kg man runs at 3 m/s for 45 min, consider the following. (a) Calculate the percentage of the power output that goes into heating up his body. (b) Calc

Question:

A person can generate about 300 W of power on a treadmill. If the treadmill is inclined at 2% and a 70-kg man runs at 3 m/s for 45 min, consider the following.

(a) Calculate the percentage of the power output that goes into heating up his body.

(b) Calculate the percentage that keeps him moving on the treadmill.

(c) How much water would that heat evaporate? Assume the water starts at body temperature, {eq}37^{circ}C {/eq}.

Answers (0)
• April 19, 2023 в 00:02
(a) The power output of the person on the treadmill is 300 W. However, not all of this power output goes towards moving the person on the treadmill. A portion of the power output will go towards heating up the person's body due to the energy required for muscle contractions and friction between the person's feet and the treadmill. The amount of power that goes towards heating up the body can be estimated using the formula: Power (in watts) = mass (in kg) x specific heat capacity of the body (in J/kg*C) x change in temperature (in C) / time (in seconds) Assuming a specific heat capacity of the body of 3500 J/kg*C and a change in temperature of 1 C, the power going towards heating up the body is: Power = 70 kg x 3500 J/kg*C x 1 C / (45 min x 60 s/min) = 27.8 W Therefore, the percentage of power output that goes towards heating up the body is: % of power output = (27.8 W / 300 W) x 100% = 9.3% (b) The remaining power output goes towards moving the person on the treadmill. The work done by the person on the treadmill can be calculated using the formula: Work (in joules) = force (in newtons) x distance (in meters) The force can be estimated using Newton's second law, which states that force = mass x acceleration. Assuming an acceleration of 0.2 m/s^2 (which corresponds to a 2% incline) and a mass of 70 kg, the force is: force = 70 kg x 0.2 m/s^2 = 14 N The distance travelled in 45 min at a speed of 3 m/s is: distance = 3 m/s x 45 min x 60 s/min = 8100 m Therefore, the work done by the person on the treadmill is: Work = 14 N x 8100 m = 113400 J The power output that goes towards moving the person on the treadmill is: Power = Work / time = 113400 J / (45 min x 60 s/min) = 42 W Therefore, the percentage of power output that keeps the person moving on the treadmill is: % of power output = (42 W / 300 W) x 100% = 14% (c) The amount of water that the heat generated by the person on the treadmill would evaporate can be calculated using the formula: Heat (in joules) = mass (in kg) x specific heat capacity of water (in J/kg*C) x change in temperature (in C) + latent heat of vapourization (in J/kg) x amount of water (in kg) Assuming 100% efficiency in the conversion of heat to evaporation, the heat generated by the person on the treadmill can be estimated by subtracting the power output going towards moving the person on the treadmill and the power going towards heating up the body from the total power output: Heat = (300 W - 42 W - 27.8 W) x 45 min x 60 s/min = 340200 J Assuming the latent heat of vapourization of water is 2257 kJ/kg and the water starts at a temperature of 37 C, the amount of water that the heat would evaporate is: amount of water = Heat / (specific heat capacity of water x change in temperature + latent heat of vapourization) amount of water = 340200 J / (4186 J/kg*C x (100 - 37) C + 2257 kJ/kg x 1000 J/kJ) = 0.50 kg or 500 g Therefore, the heat generated by the person on the treadmill would evaporate 500 g of water.
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