The head of a grass string trimmer has 100 g of cord wound in a light, cylindric
ID: 2298035 • Letter: T
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 0.200 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 2 040 rev/min and the grass exerts an average tangential force of 7.10 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
) Average power = total energy imparted / time
Total energy = rotational + translational energy
Rotational energy = 1/2 inertia * ?^2
Inertia, I = inertia of wounded cord + extended cord
This involves inertia of annulus + rod using parallel axis theorem
I = 4.659e-4 kg m^2
(You verify this)
? = 2700*2?/60
E_rot = 1/2 * I *?^2 = 18.62 J
The extended cord also has linear translation
v = ? * 0.17
E_tr = 1/2 m v^2 = 1.85 J
E_total = 20.47 J
Average power = 20.47 / 0.185 = 110.66 W
b) Power = torque x angular speed
Torque = 8.75 * 0.25 Nm
Power = 8.75 * 0.25 * 2160 * 2?/60
= 494.8 W
*EDIT*
I initially made a mistake in part a) which I corrected. I believe your book does not account for the translational energy of the extended cord. If I use only the 18.62 J of rotational energy divided by time, I will get 100.7 J. Also it is not clear whether the mass of 100 g includes the extended cord or not. This may account for the discrepancy between my answer and the book's.
*EDIT2*
OK I redid the calculations using 98.4 g of cord wound up and 1.6 g of cord extended.
Total inertia = 4592.43 g cm^2
? = 2700*2?/60
E_rot = 18.36 J
Dividing this by 0.185 will give 99.2 W.
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