1. An object moves along a straight track from the point (1,−2,1) to the point (0,−7,5). The only force acting on it is a constant F=2i−3j−5k newtons. Find the work done if the distance is measured in meters.
2. A wrench 0.5 meters long lies along the positive y-axis, and grips a bolt at the origin. A force is applied in the direction of <0,1,4> at the end of the wrench. Find the magnitude of the force in newtons needed to supply 100 newton-meters of torque to the bolt.
since F is conservative, the work is independent of the path, so it is
∫[(1,-2,1),(0,-7,5)] 2dx-3dy-5dz
= 2x-3y-5z [(1,-2,1),(0,-7,5)]
= (0+21-25)-(2+6-5)
= -4-3
= -7
You can work the other the same way, but it's easier since only y is involved.
To find the work done in question 1, we can use the formula for work done by a force:
W = F ∙ d
where W is the work done, F is the force, and d is the displacement vector.
1. Calculate the displacement vector:
The displacement vector, d, can be found by subtracting the initial position vector from the final position vector:
d = (0 - 1)i + (-7 - (-2))j + (5 - 1)k
= -i - 5j + 4k
2. Calculate the magnitude of the force:
To calculate the magnitude of the force, we use the formula:
|F| = sqrt(Fx^2 + Fy^2 + Fz^2)
where Fx, Fy, and Fz are the components of the force vector.
|F| = sqrt((2)^2 + (-3)^2 + (-5)^2)
= sqrt(4 + 9 + 25)
= sqrt(38)
3. Calculate the dot product of the force and displacement vectors:
The dot product of two vectors can be calculated by multiplying their corresponding components and summing them:
F · d = (2)(-1) + (-3)(-5) + (-5)(4)
= -2 + 15 - 20
= -7
4. Calculate the work done:
Using the formula for work done, we can substitute the values:
W = (-7)m
= -7 joules
Therefore, the work done is -7 joules.
For question 2, we need to find the magnitude of the force needed to supply a certain torque to the bolt:
1. Calculate the cross product:
To find the torque, we need to take the cross product of the position vector and the applied force vector. The cross product of two vectors can be calculated using the formula:
A x B = (AyBz - AzBy)i + (AzBx - AxBz)j + (AxBy - AyBx)k
In this case, the position vector A is (0, -0.5, 0) and the applied force vector B is (0, 1, 4). Calculating the cross product:
A x B = (0)(4) - (0)(1)i + (0)(0) - (0)(4)j + (-0.5)(1) - (0)(0)k
= 0i + 0j - 0.5k
= -0.5k
2. Calculate the magnitude of the torque:
The magnitude of the torque can be found using the formula:
|τ| = |A||B|sinθ
where |A| and |B| are the magnitudes of the vectors and θ is the angle between them. In this case, |A| = 0.5 meters and |B| = √(0^2 + 1^2 + 4^2) = √17 meters.
|τ| = (0.5)(√17)sin90°
= 0.5√17
3. Calculate the magnitude of the force:
To supply 100 newton-meters of torque, we use the formula:
|τ| = |F||r|sinθ
where |F| is the magnitude of the force, |r| is the length of the wrench, and θ is the angle between the force and the wrench.
0.5√17 = |F|(0.5)sin90°
|F| = (0.5√17)/(0.5)
= √17
Therefore, the magnitude of the force needed to supply 100 newton-meters of torque to the bolt is √17 newtons.