Threepersons 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?
note=figure values= F1= 45N, 70degree , F2= 75N,20degree
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.
(a) First, let's resolve the forces F1 and F2 into their horizontal and vertical components.
For F1:
Horizontal Component: F1x = F1 * cos(70°)
Vertical Component: F1y = F1 * sin(70°)
For F2:
Horizontal Component: F2x = F2 * cos(20°)
Vertical Component: F2y = F2 * sin(20°)
Next, let's find the resultant horizontal and vertical forces acting on the cart.
Horizontal Resultant Force: Fx = F1x + F2x
Vertical Resultant Force: Fy = F1y + F2y
Since we want to stop the cart, the third person's force should be equal and opposite to the resultant force. Therefore, the magnitude and direction of the force the third person should exert are given by:
Magnitude: |F3| = √(Fx² + Fy²)
Direction: θ = arctan(Fy/Fx)
Now, let's calculate the values using the given information:
F1 = 45N, 70°
F2 = 75N, 20°
F1x = 45 * cos(70°) ≈ 15.00 N
F1y = 45 * sin(70°) ≈ 40.95 N
F2x = 75 * cos(20°) ≈ 70.67 N
F2y = 75 * sin(20°) ≈ 25.77 N
Fx = F1x + F2x ≈ 15.00 N + 70.67 N ≈ 85.67 N
Fy = F1y + F2y ≈ 40.95 N + 25.77 N ≈ 66.72 N
|F3| = √(85.67² + 66.72²) ≈ 108.78 N
θ = arctan(66.72 N / 85.67 N) ≈ 38.09°
Therefore, the magnitude of the force that the third person should exert to stop the cart is approximately 108.78 N, and the direction is approximately 38.09°.
(b) To find the weight of the cart, we can use the acceleration of the cart and the following equation:
Net Force = Mass * Acceleration
The net force acting on the cart is the resultant force (Fx) minus the force exerted by the third person (-F3). So:
Net Force = Fx - F3
Mass * Acceleration = Fx - F3
We know the acceleration is 200 m/s². Rearranging the equation, we can solve for the mass of the cart:
Mass = (Fx - F3) / Acceleration
Now, let's substitute the values we have:
Fx = 85.67 N
F3 = 108.78 N
Acceleration = 200 m/s²
Mass = (85.67 N - 108.78 N) / 200 m/s² ≈ -0.1155 kg
Since mass cannot be negative, there might be an error in the calculations or the given data. Please double-check the values and calculations to ensure accuracy.