Calculating Total Weight of a Truck with Uneven Loads

[tandoorichicken]

A trucker needs to weigh a truck that is too long to fit on a platform scale. When the front wheels of the truck are run onto the scale, the scale reads $$W_1$$. When the rear wheels are run onto the scale so that the front wheels are off, it reads $$W_2$$.
a) prove that the total wight of the truck is $$W_1 + W_2$$.
b) prove that if the truck is loaded so that its center of gravity is halfway between the front and rear wheels, the total weight is $$2W_1$$.

 

[ShawnD]

Originally posted by tandoorichicken
A trucker needs to weigh a truck that is too long to fit on a platform scale. When the front wheels of the truck are run onto the scale, the scale reads $$W_1$$. When the rear wheels are run onto the scale so that the front wheels are off, it reads $$W_2$$.
a) prove that the total wight of the truck is $$W_1 + W_2$$.
b) prove that if the truck is loaded so that its center of gravity is halfway between the front and rear wheels, the total weight is $$2W_1$$.

Here is my FBD for case 1, the scale is on the left.

force up (W1)------x distance-------centre of grav (G)---d distance----wheels (A)

The moment around A looks like this

$$0 = dG - W_1(d + x)$$ no movement

$$W_1(d + x) = dG$$ isolating for W1

$$W_1 = \frac{dG}{d + x}$$ isolating for W_1

Now here is case 2 where the scale is the on the right.

wheels (A)-----x distance-------centre of grav (G)---d distance-----force up (W2)

Moment around the new A

$$0 = W_2(d + x) - xG$$ no movement

$$W_2(d + x) = xG$$ isolating W_2

$$W_2 = \frac{xG}{d + x}$$ isolating W_2



Now you are supposed to prove that W1 + W2 = G so just fill in now

$$W_1 + W_2 = G$$ simple version

$$\frac{dG}{d + x} + \frac{xG}{d + x} = G$$ expanded version

$$\frac{dG + xG}{d + x} = \frac{G(d + x)}{d + x}$$ common denominator

$$\frac{dG + xG}{d + x} = \frac{dG + xG}{d + x}$$ expanded

They are the same are they not? :D


Once you solve A, B is very easy. If G is to equal 2W1, all you need to prove is that W2 and W1 are equal. If the centre of gravity is in the middle, the equations for W1 and W2 will be the exact same.

 

[M98Ranger]

a) To prove that the total weight of the truck is W_1 + W_2, we can use the principle of moments. The principle of moments states that the sum of the moments acting on a body is equal to the moment of the resultant force acting on that body.

In this case, the moment of the resultant force acting on the truck is equal to the weight of the truck multiplied by the distance between the front and rear wheels. So, we have:

Moment of resultant force = Weight of truck x Distance between front and rear wheels

Since the weight of the truck is distributed between the front and rear wheels, we can write:

Weight of truck = W_1 + W_2

And since the distance between the front and rear wheels is the same for both W_1 and W_2, we can write:

Distance between front and rear wheels = d

Substituting these values into the equation for the moment of resultant force, we get:

Moment of resultant force = (W_1 + W_2) x d

Now, using the principle of moments, we can equate this to the sum of the moments acting on the truck:

(W_1 x d) + (W_2 x d) = (W_1 + W_2) x d

Simplifying, we get:

W_1 x d + W_2 x d = W_1 x d + W_2 x d

Therefore, the total weight of the truck is W_1 + W_2.

b) To prove that if the truck is loaded so that its center of gravity is halfway between the front and rear wheels, the total weight is 2W_1, we can use the same principle of moments.

In this case, the moment of resultant force acting on the truck is equal to the weight of the truck multiplied by the distance between the center of gravity and the front wheels. So, we have:

Moment of resultant force = Weight of truck x Distance between center of gravity and front wheels

Since the center of gravity is halfway between the front and rear wheels, the distance between the center of gravity and the front wheels is equal to half the distance between the front and rear wheels. So, we can write:

Distance between center of gravity and front wheels = d/2

Substituting this into the equation for the moment of resultant force, we get:

Moment of resultant force = Weight of truck x (d