Physics, 26.10.2019 00:43 jennemylesp19oy5

The head of a grass string trimmer has 100 g of cord wound in a light, cylindrical spool with inside diameter 3.00 cm and outside diameter 18.0 cm as shown in the figure below. the cord has a linear density of 10.0 g/m. a single strand of the cord extends 16.0 cm from the outer edge of the spool. (a) when switched on, the trimmer speeds up from 0 to 2 250 rev/min in 0.230 s. what average power is delivered to the head by the trimmer motor while it is accelerating? w (b) when the trimmer is cutting grass, it spins at 1 915 rev/min and the grass exerts an average tangential force of 8.80 n on the outer end of the cord, which is still at a radial distance of 16.0 cm from the outer edge of the spool. what is the power delivered to the head under load?

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Answer from: iBrain

The rotational equivalence of a force is a torque

(a) Power delivered to the head is approximately 56.23 W

(b) Power delivered to the head under load is approximately 441.184 W

The reason why the above values are correct is as follows;

The known parameters are;

Mass of the cord, m = 100 g

Inside diameter of the cord, D₁ = 3.00 cm (∴ r₁ = 0.015 m)

Outside diameter of the cord, D₂ = 18.00 cm (∴ r₂ = 0.09 m)

Linear density of the cord = 10.0 g/m

The length of a strand of cord, L = 16.0 cm

(a) Time it takes the trimmer to speed up to 2,250 rev/min from 0 is 0.230 s.

Required;

The average power delivered to the head by the trimmer motor while it is accelerating

Solution;

$The \ average \ power =\dfrac{Rotational \ kinetic \ energy}{Time}$

The rotational kinetic energy = $E_{rotational } = \dfrac{1}{2} \cdot I \cdot \omega^2$

$\omega = \dfrac{ 2,250 \times 2\times \pi}{60} = 75$

Angular speed, ω = 75 rad/s

I = Moment of inertia of the cord + Moment of inertia of strand

The cord is a hollow cylinder

Moment of inertia, I, of a hollow cylinder, with radii, a and b, is given as follows;

$I = \dfrac{1}{2} \cdot M \cdot (a^2 + b^2)$

Therefore, for the cord, we have;

$I = \dfrac{1}{2} \cdot M \cdot (r_1^2 + r_2^2)$

Therefore, we have;

$I_{cord} = \dfrac{1}{2} \times 0.1 \ kg\times ((0.09 \, m)^2 + (0.015 \, m)^2) = 0.00041625 \ kg/m^2$

$I_{cord}$ = 4.1625 × 10⁻⁴ kg/m²

The moment of inertia of the strand that extends from the cord is given as follows;

$I_{strand} = I_{cm} + M\cdot D^2$

Moment of inertia of cord

$I_{strand} = \dfrac{m \cdot L^2}{12} + m\cdot \left(\dfrac{L}{2} + r_2 \right)^2$

$I_{strand} = m \cdot \left(\dfrac{L^2}{12} + \left(\dfrac{L}{2} + r_2 \right)^2\right)$

$I_{strand} = \dfrac{10 \times 0.16}{1,000} \times \left(\dfrac{0.16^2}{12} + \left(\dfrac{0.16}{2} + 0.09 \right)^2\right) = \dfrac{115}{2316058} = 4.965\bar {3} \times 10^{-5}$

$I_{strand}$ = 4.965 $\bar 3$ kg/m²

The total moment of inertia, I = $I_{cord}$ + $I_{strand}$

∴ I = 4.1625 × 10⁻⁴ kg/m² + 4.965 $\bar 3$ kg/m² = 4.659 × 10⁻⁴ kg/m²

$E_{rotational } = \dfrac{1}{2} \times 4.659 \times 10^{-4}\cdot (75\times \pi)^2 \approx 12.933 \ J$

$Power = \dfrac{Energy}{Time}$

$Power\ delivered \ to \ head \approx \dfrac{12.933 \ J}{0.230 \ s} = 56.23 \ W$

(b) The rotational speed under load = 1,915 rev/min

The tangential force exerted on the outer end of the cord, F = 8.80 N

The distance of the cord from the spool = 16.0 cm

Torque = F × d

d = 0.16 m + 0.09 m = 0.25 m

Torque = 8.80 N × 0.25 m = 2.2 N·m

Power = Torque × ω

$\omega = \dfrac{ 1,915 \times 2\times \pi}{60} \approx 200.583$

ω ≈ 200.583 rad/s

Power = 2.2 N·m × 200.583 rad ≈ 441.184 W

Power delivered to the head under load ≈ 441.184 W

Learn more about rolling torque here:

link

Answer from: Quest

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Predict how the weather in chicago, il will change in the next 12 hours. a. it will experience clea

Answer from: Quest

as we know our atmosphere or air have different levels. every level has other gasses, and every gas has another permeability. when the light comes from lower permeability to higher permeability, it changes its direction. so that means that when the light comes from the moon to the earth, it changes it's direction many more times. it basically means that the shape of the light part of the moon is changed on the earth.

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