Three horizontal forces (the only horizontal one) act on a box sitting on a floor. One force (F1) acts due east and has a magnitude of 150 lb. A second force (F2) has an easterly component at 30.0 lb and a southerly component at 40.0 lb. The box remains at rest. Sketch the two known forces on the box. Find the unknown third force in newtons.
The figure shows two forces acting at right angles on an object. They have magnitudes F1 = 6.3 N and F2 = 2.1 N. What third force will cause the object to be in equilibrium (acceleration equals zero)?
To sketch the known forces on the box, we can use a vector diagram. First, draw a horizontal line to represent the floor. Then, draw a vector pointing to the right (east) with a length of 150 lb to represent F1. Finally, draw a vector with an easterly component of 30.0 lb and a southerly component of 40.0 lb to represent F2.
F2
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F1 Floor
To find the unknown third force (F3) in newtons, we need to consider the equilibrium of forces. Since the box remains at rest, the sum of the horizontal forces must be zero.
The horizontal component of F2 cancels the horizontal component of F1, so:
F2_horizontal = F1 = 150 lb
Now, we need to convert the forces to newtons. 1 pound (lb) is approximately equal to 4.45 newtons (N).
F1 = 150 lb x 4.45 N/lb ≈ 667.5 N
F2_horizontal = 150 lb x 4.45 N/lb ≈ 667.5 N
Therefore, the magnitude of the unknown third force (F3) is 667.5 N.
To begin, let's sketch the two known forces on the box.
The first force, F1, acts due east with a magnitude of 150 lb. Thus, we draw an arrow pointing to the right (eastward) with a length representing the magnitude of 150 lb.
The second force, F2, has an easterly component of 30.0 lb and a southerly component of 40.0 lb. To represent this force, we draw two arrows: one pointing to the right with a length representing the easterly component of 30.0 lb, and another arrow pointing downwards (southward) with a length representing the southerly component of 40.0 lb. These arrows should start from the same point as F1.
Now, since the box remains at rest, the net force acting on it must be zero in both the horizontal and vertical directions. This means the two known forces (F1 and F2) must be balanced by the unknown third force (F3) in order to keep the box in equilibrium.
Let's find the unknown third force (F3) in newtons.
First, let's convert the given forces from pounds (lb) to newtons (N), since we need consistent units for our calculations. Recall that 1 lb is approximately equal to 4.448 N.
F1 = 150 lb * 4.448 N/lb = 667.2 N (approximately)
F2 (easterly component) = 30.0 lb * 4.448 N/lb = 133.44 N (approximately)
F2 (southerly component) = 40.0 lb * 4.448 N/lb = 177.92 N (approximately)
Now, since the net horizontal force acting on the box is zero, the magnitudes of the easterly components of F1 and F2 must be equal.
Thus, F3 = F1 - F2 (easterly component)
F3 = 667.2 N - 133.44 N = 533.76 N (approximately)
Therefore, the unknown third force, F3, has a magnitude of approximately 533.76 N.