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.

To find the weight on the nearer wheels of the cart under this load, we need to consider the distribution of weight due to the crate.

The center of mass of the crate is located 0.525 m from the center of the cart. Since the crate has a mass of 109 kg, we can calculate the weight of the crate as W_crate = mass_crate * gravity, where gravity is approximately 9.8 m/s^2.

W_crate = 109 kg * 9.8 m/s^2 = 1068.2 N

Now, since the crate is located in the back-middle of the cart, we can assume that the weight of the crate acts on the center of the cart. Therefore, the weight of the crate is evenly distributed between the four wheels of the cart.

To calculate the weight on each wheel, we divide the weight of the crate evenly between the four wheels. So, the weight on each of the nearer wheels is:

W_nearer = W_crate / 2 = 1068.2 N / 2 = 534.1 N

Therefore, the weight on each of the nearer wheels of the cart under this load is 534.1 N.

To find the weight on the farther wheels of the cart under this load, we use the fact that the distance between a wheel and its nearest neighbors is 2.1 m. Since the crate is placed 0.525 m from the center of the cart, it means that the weight of the crate is distributed asymmetrically between the nearer and farther wheels.

To calculate the weight on each of the farther wheels, we consider the moment of the weight distribution caused by the crate. The moment is given by the weight multiplied by the distance from the center of the cart.

The moment on each side is the same due to the symmetry of the cart. So, we can calculate the moment caused by the crate on one of the farther wheels as follows:

Weight_moment = W_crate * distance

Weight_moment = 1068.2 N * 0.525 m = 559.755 Nm

Since there are two farther wheels, we divide the moment by two to find the weight on each of the farther wheels:

Weight_on_each_farthest_wheel = Weight_moment / distance_between_wheels

Weight_on_each_farthest_wheel = 559.755 Nm / 2.1 m = 266.5 N

Therefore, the weight on each of the farther wheels of the cart under this load is 266.5 N.

To solve this problem, we can start by calculating the weight of the entire system and then distributing it between the nearer and farther wheels.

Given:
Mass of the cart (m_cart) = 11 kg
Distance between a wheel and its nearest neighbors (d) = 2.1 m
Mass of the crate (m_crate) = 109 kg
Center of mass of the crate from the center of the cart (d_center) = 0.525 m

a) To find the weight on the nearer wheels of the cart, we need to calculate the weight of the crate that is acting on the nearer wheels. Since the center of mass of the crate is located 0.525 m from the center of the cart, the weight acting on the nearer wheels is given by:

Weight on nearer wheels = Weight of the crate * (Distance to nearer wheels / Distance between wheels)

Weight on nearer wheels = m_crate * (d_center / d)
Weight on nearer wheels = 109 kg * (0.525 m / 2.1 m)

b) Similarly, to find the weight on the farther wheels of the cart, we need to calculate the weight of the crate that is acting on the farther wheels. Since the center of mass of the crate is located 0.525 m from the center of the cart, the weight acting on the farther wheels is given by:

Weight on farther wheels = Weight of the crate * (Distance to farther wheels / Distance between wheels)

Weight on farther wheels = m_crate * (d_center / d)
Weight on farther wheels = 109 kg * (0.525 m / 2.1 m)

Now, we can calculate the values:

a) Weight on nearer wheels = 109 kg * (0.525 m / 2.1 m) = 27.25 kg
b) Weight on farther wheels = 109 kg * (0.525 m / 2.1 m) = 27.25 kg

So, the weight on both the nearer and farther wheels of the cart under this load is 27.25 kg each.