some recent calculations based on real numbers
over a second it takes a deep breath to spin the 5 g pen for 240 rpm, without the weight I get 480 rpm from the same breath. what is the Nm of the torque?
Assistant
Poe
To calculate the torque produced by the toothpick while spinning the
5 g pen at
240 RPM and to compare it with the
480 RPM without the weight, we can follow these steps:
Given Information
- Weight of the Pen: 5 grams (0.005 kg)
- Rotational Speed with Pen: 240 RPM
- Rotational Speed without Pen: 480 RPM
- Duration for Deep Breath: 1 second
- Energy from One Breath: Assuming 0.1 joules, therefore the energy used for the pen is 0.1 joules.
Steps to Calculate Torque
1. Calculate the Force
The force due to the weight of the pen can be calculated as follows:
F=m×gF = m \times gF=m×g
Where:
- m=0.005 kgm = 0.005 \, \text{kg}m=0.005kg
- g≈9.81 m/s2g \approx 9.81 \, \text{m/s}^2g≈9.81m/s2
Calculating the force:
F=0.005 kg×9.81 m/s2≈0.04905 NF = 0.005 \, \text{kg} \times 9.81 \, \text{m/s}^2 \approx 0.04905 \, \text{N}F=0.005kg×9.81m/s2≈0.04905N
2. Calculate Angular Velocity
Convert the RPM to radians per second for the pen:
ω=240 RPM×2π60≈25.13 rad/s\omega = 240 \, \text{RPM} \times \frac{2\pi}{60} \approx 25.13 \, \text{rad/s}ω=240RPM×602π≈25.13rad/s
3. Determine the Effective Radius
Assuming the pen is 0.15 m long and attached at its midpoint:
r=0.15 m2=0.075 mr = \frac{0.15 \, \text{m}}{2} = 0.075 \, \text{m}r=20.15m=0.075m
4. Calculate Torque
Using the torque formula:
τ=F×r\tau = F \times rτ=F×r
Substituting in the values:
τ=0.04905 N×0.075 m\tau = 0.04905 \, \text{N} \times 0.075 \, \text{m}τ=0.04905N×0.075m
Calculating the torque:
τ≈0.00367875 Nm\tau \approx 0.00367875 \, \text{Nm}τ≈0.00367875Nm
Conclusion
The estimated torque produced by the toothpick while spinning the
5 g pen at
240 RPM is approximately
0.00368 Nm (or 3.68 mNm).
If you would like to compare this with the torque when spinning without the weight, we can calculate that as well!
what Is the estimated efficiency at that torque?
Assistant
Poe
To calculate the efficiency at the torque of
0.00368 Nm when spinning the
5 g pen at
240 RPM, we need to compare the output power (based on the torque) to the input power (based on the energy from the breath).
Given Information
- Torque: τ=0.00368 Nm\tau = 0.00368 \, \text{Nm}τ=0.00368Nm
- Rotational Speed with Pen: 240 RPM240 \, \text{RPM}240RPM
- Energy from One Breath: Assume 0.1 joules.
- Duration for Deep Breath: 1 second1 \, \text{second}1second
Steps to Calculate Efficiency
1. Calculate Input Power
Using the energy input over the duration:
Input Power=EnergyTime=0.1 J1 s=0.1 W\text{Input Power} = \frac{\text{Energy}}{\text{Time}} = \frac{0.1 \, \text{J}}{1 \, \text{s}} = 0.1 \, \text{W}Input Power=TimeEnergy=1s0.1J=0.1W
2. Calculate Angular Velocity
Convert the RPM to radians per second:
ω=240 RPM×2π60≈25.13 rad/s\omega = 240 \, \text{RPM} \times \frac{2\pi}{60} \approx 25.13 \, \text{rad/s}ω=240RPM×602π≈25.13rad/s
3. Calculate Output Power
Using the torque and angular velocity:
Output Power=τ×ω\text{Output Power} = \tau \times \omegaOutput Power=τ×ω
Substituting in the values:
Output Power=0.00368 Nm×25.13 rad/s≈0.0927 W\text{Output Power} = 0.00368 \, \text{Nm} \times 25.13 \, \text{rad/s} \approx 0.0927 \, \text{W}Output Power=0.00368Nm×25.13rad/s≈0.0927W
4. Calculate Efficiency
Now, we can calculate the efficiency:
Efficiency(%)=(Output PowerInput Power)×100\text{Efficiency} (\%) = \left( \frac{\text{Output Power}}{\text{Input Power}} \right) \times 100Efficiency(%)=(Input PowerOutput Power)×100
Substituting the values:
Efficiency=(0.0927 W0.1 W)×100≈92.7%\text{Efficiency} = \left( \frac{0.0927 \, \text{W}}{0.1 \, \text{W}} \right) \times 100 \approx 92.7\%Efficiency=(0.1W0.0927W)×100≈92.7%
Conclusion
The estimated efficiency at the torque of
0.00368 Nm while spinning the
5 g pen at
240 RPM is approximately
92.7%. This indicates that a significant portion of the input energy is being converted into useful work, which is a good level of efficiency!
If you have any further questions or need additional calculations, feel free to ask!