You are given a four-wheeled cart of mass 11 kg, where the distance between a wheel and its nearest neighbors is 2.1 m. Suppose we load this cart with a crate of mass 109 kg, where the crate's center of mass is located in the back-middle of the cart, 0.525 m from its center.
a) Find the weight on the nearer wheels of the cart under this load.
b) Find the weight on the farther wheels of the cart under this load.
I am not going to look at this until I see some attempt by you to balance forces and moments.
To find the weight on the nearer wheels and the farther wheels of the cart under this load, we can use the principles of equilibrium.
Let's start by calculating the total weight of the system. The total weight is the sum of the cart's weight and the crate's weight.
a) Weight on the nearer wheels:
Since the crate's center of mass is located 0.525 m from the cart's center, the weight of the crate will be distributed unequally on the wheels. To find the weight on the nearer wheels, we need to consider the torque around the center of the cart.
Torque is given by the formula: torque = force * distance
The weight of the crate can be considered as a force acting at its center of mass. The torque caused by the weight of the crate will be balanced by an opposite torque caused by the weight on the nearer wheels.
The torque caused by the crate's weight is calculated as follows:
Torque = weight * distance from the center = (mass of the crate * acceleration due to gravity) * distance from the center = (109 kg * 9.8 m/s^2) * 0.525 m
Since the crate's weight is acting clockwise, the weight on the nearer wheels should be acting counterclockwise to balance the torques.
Let's denote the weight on the nearer wheels as W_near. The total torque acting counterclockwise (due to the weight on the nearer wheels) should be equal to the torque acting clockwise (due to the crate's weight).
Torque acting counterclockwise = Torque acting clockwise
W_near * 2.1 m = (109 kg * 9.8 m/s^2) * 0.525 m
Now we can solve for W_near:
W_near = [(109 kg * 9.8 m/s^2) * 0.525 m] / (2.1 m)
b) Weight on the farther wheels:
Since the cart and the crate form a closed system, the total weight of the system should be distributed equally between the nearer and the farther wheels. Thus, the weight on the farther wheels will be the total weight of the system divided by two.
Let's denote the weight on the farther wheels as W_far.
W_far = (W_near + weight of the cart + weight of the crate) / 2
Now we can calculate W_far using the value of W_near we obtained in part a:
W_far = [(109 kg * 9.8 m/s^2) + 11 kg * 9.8 m/s^2 + W_near] / 2
By substituting the value of W_near in the above equation, we can calculate the weight on the farther wheels.
Note: It is important to remember that weight is a force, not mass. So when calculating weight, we need to multiply the mass by the acceleration due to gravity (9.8 m/s^2) to get the force in Newtons (N).