The head of a grass string trimmer has 100 g of cord wound in a light, cylindric

Question

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 2 550 rev/min in 0.175 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 2 060 rev/min and the grass exerts an average tangential force of 6.55 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?
W

Explanation / Answer

the moment of inertia of the cord on the spool is

1/2 M(R1^1 + R2^2) = 1/2 * (0.100 ) [( 0.015)^2 + (0.090 )^2]

I = 4.16 x 10^-4 kg . m^2

stand mass = 10 x 10^-2 x .160 = 1.6 x 10^-3 kg

and moment of inertia

I = Icm + Md^2 = 1/12 * ML^2 + Md^2

= 4.97 x 10^-5 kg .m^2

for the whole cord = I = 4.66 x 10^-4 kg .m^2

P = E/t

E = 1/2 * Iw^2

w = 2550 rev/min = 267.035 rad/s

P = 94.94 W

par b )

P = torque * w

P = F*r *w

P =6.55 * ( 0.160 + 0.090) * (2060 *2pi/60)

P = 353.25 W

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