1m long spade balances in a string loop when the loop is 30 cm from the end of the blade. A 2 kg mass is placed at the handle and the balance point is the centre of the spade. What is the mass of the spade?
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Please see the diagram.
Mass of blade = m kg. Mass of handle is M. kg Mass of spade = M+m. kg
We balance the moments of mass/weights, using products of arm lengths to the centers of masses and masses.
from the top figure,
[ (30/L)m ] * 15 = [(L-30)/L * m ] * (L-30)/2 + M * (20 + L/2)
900*m = m (L-30)² + M *(40+L) L
60 m = 40 M + (m+M) L --- (1 )
from the next diagram, we get:
(50 - L/2) m + (50-L)/2 * [(50-L)*M/(100-L)] = 2 *50 + 25 * [ 50*M/(100-L) ]
so we get,
(100-L) [100 m - L m - 200 ] = M (100 -L ) L
100 m = 200 + (m+M) L
using --(1) we get
100 m = 200 + 60 m -40 M
40 (M+m) = 200
M+m = total weight of spade = 200/40 = 5 kg
Mass of blade = m kg. Mass of handle is M. kg Mass of spade = M+m. kg
We balance the moments of mass/weights, using products of arm lengths to the centers of masses and masses.
from the top figure,
[ (30/L)m ] * 15 = [(L-30)/L * m ] * (L-30)/2 + M * (20 + L/2)
900*m = m (L-30)² + M *(40+L) L
60 m = 40 M + (m+M) L --- (1 )
from the next diagram, we get:
(50 - L/2) m + (50-L)/2 * [(50-L)*M/(100-L)] = 2 *50 + 25 * [ 50*M/(100-L) ]
so we get,
(100-L) [100 m - L m - 200 ] = M (100 -L ) L
100 m = 200 + (m+M) L
using --(1) we get
100 m = 200 + 60 m -40 M
40 (M+m) = 200
M+m = total weight of spade = 200/40 = 5 kg
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