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.
(b) If the third person exerts the force found in part (a), the cart accelerates at 200 m/S2 in the
(+) x-direction. What is the weight of the cart?
No figure. Cannot copy and paste here.
To find the magnitude and direction of the force that the third person should exert to stop the cart, we need to consider the forces acting on the cart.
(a) Let's assume that the person on the left is pushing with a force F1, and the person on the right is pushing with a force F2. The force exerted by the third person can be represented as F3.
Since the cart is not accelerating vertically, the vertical forces must be balanced. Therefore, the vertical components of the pushing forces must cancel out the weight of the cart.
The weight of the cart is given by the equation W = m * g, where m is the mass of the cart and g is the acceleration due to gravity.
Assuming the pushing forces are applied horizontally, the vertical components of forces F1 and F2 would be zero. Therefore, the vertical component of the force exerted by the third person, F3, must be equal to the weight of the cart, W.
Hence, we have F3 = W = m * g.
(b) If the third person exerts the force found in part (a), and the cart accelerates at 200 m/s^2 in the positive x-direction, we can apply Newton's second law of motion, which states that the net force on an object is equal to the mass of the object multiplied by its acceleration.
In this case, the net force acting on the cart is the sum of the horizontal forces exerted by all three people: F_net = F1 + F2 + F3.
Since the cart is accelerating in the x-direction, the net force must be equal to the mass of the cart multiplied by its acceleration: F_net = m * a.
Combining both equations, we get:
F1 + F2 + F3 = m * a
Given that the cart accelerates at 200 m/s^2, we can substitute this value into the equation:
F1 + F2 + F3 = m * 200
To find the weight of the cart (W), we substitute F3 = W as found in part (a):
F1 + F2 + W = m * 200
Now, we have an equation with three unknowns (F1, F2, and W), but we can solve for one of them because we know that F1 and F2 are horizontal forces. They cancel out each other vertically, so their vertical components must be equal in magnitude.
By applying the law of cosines, the magnitude of F3 is found to be:
|F3| = sqrt(F1^2 + F2^2 + 2 * F1 * F2 * cos(theta))
where theta is the angle between F1 and F2.
Using this magnitude equation and the fact that F3 = W, we can solve for W.
In summary, to find the magnitude and direction of the force that the third person should exert to stop the cart, we need to calculate the weight of the cart using the equation W = m * g, substitute it into the equation F1 + F2 + W = m * 200, and solve for W.