A machine carries a 5.0 kg package from an initial position of di = (0.50 m) i + (0.75 m) j + (0.20 m) k at t = 0 to a final position of df = (8.00 m) i + (11.0 m) j + (7.40 m) k at t = 13 s. The constant force applied by the machine on the package is F = (2.00 N) i + (4.00 N) j + (6.00 N) k.
(a) For that displacement, find the work done on the package by the machine's force.
(b) Also, find the power of the machine's force on the package.
Work= Force*displacement
Power= Work/time
To calculate the work done on the package by the machine's force, we can use the formula:
Work = Force * Displacement
In this case, the force applied by the machine is given as F = (2.00 N) i + (4.00 N) j + (6.00 N) k, and the displacement of the package is given as df = (8.00 m) i + (11.0 m) j + (7.40 m) k - di = (8.00 - 0.50) m i + (11.0 - 0.75) m j + (7.40 - 0.20) m k.
Let's calculate the work done:
Work = (2.00 N) i + (4.00 N) j + (6.00 N) k * ((8.00 - 0.50) m i + (11.0 - 0.75) m j + (7.40 - 0.20) m k)
Now, we can evaluate the dot product of the force and displacement vectors:
Work = (2.00 N)(8.00 - 0.50) m + (4.00 N)(11.0 - 0.75) m + (6.00 N)(7.40 - 0.20) m
Work = 15.5 J + 41.0 J + 43.2 J
Work = 99.7 J
Therefore, the work done on the package by the machine's force is 99.7 J.
To calculate the power of the machine's force on the package, we can use the formula:
Power = Work / Time
In this case, the time is given as t = 13 s.
Let's calculate the power:
Power = 99.7 J / 13 s
Power ≈ 7.67 W
Therefore, the power of the machine's force on the package is approximately 7.67 watts.