Q # 6: Three persons wants to push a wheel cart in the direction marked x in Fig.
The two person push with horizontal forces F1 and F2 as shown in the figure.
(a) Find the magnitude and direction of the force that third person should exert to stop this cart. You can ignore the effects of friction.
To find the magnitude and direction of the force that the third person should exert to stop the cart, we need to analyze the forces acting on the cart.
In this scenario, the two persons are pushing the cart with horizontal forces F1 and F2. To stop the cart, the third person needs to exert a force in the opposite direction.
Let's break down the forces acting on the cart:
1. F1: The force exerted by the first person. Since the direction of this force is to the right in the x-axis, we can write it as F1 = F1i, where i represents the x-axis unit vector.
2. F2: The force exerted by the second person. Again, this force is in the positive x-axis direction, so we can express it as F2 = F2i.
3. F3: The force exerted by the third person to stop the cart. Since the third person wants to stop the cart, the force should be in the direction opposite to the motion. Therefore, we can write F3 = -F3i.
According to Newton's second law, the net force acting on an object is equal to the mass of the object multiplied by its acceleration:
ΣF = ma
In this case, since the cart is at rest, the net force on the cart is zero:
ΣF = 0
So we can write:
F1 + F2 + F3 = 0
Substituting the expressions for F1, F2, and F3:
F1i + F2i + (-F3)i = 0
Adding up the vectors:
(F1 + F2 - F3)i = 0
Since the components in the i direction must add up to zero, we can write:
F1 + F2 - F3 = 0
Now, we can solve for F3:
F3 = F1 + F2
Therefore, the magnitude of the force that the third person should exert to stop the cart is equal to the sum of the magnitudes of the forces exerted by the first and second person.
As for the direction, since the forces exerted by the first and second person are in the positive x-axis direction, the force exerted by the third person should be in the negative x-axis direction to stop the cart.