As depicted in Fig. 5-13, two people sit in a car that...

As depicted in Fig. $5-13$, two people sit in a car that weighs $8000 \mathrm{~N}$. The person in front weighs $700 \mathrm{~N}$, while the one in the back weighs 900 N. Call $L$ the distance between the front and back wheels. The car's center of gravity is a distance $0.400 L$ behind the front wheels. How much force does each front wheel and each back wheel support if the people are seated along the centerline of the car?

Key Concepts

Static Equilibrium

This concept refers to a state where all forces and moments in a system are balanced, meaning the net force and net torque are both zero. It is essential for analyzing structures and mechanical systems to ensure they remain at rest under various loads.

Torque and Moment Balance

Torque is the product of a force and its lever arm, which determines the rotational influence of the force around a pivot point. Moment balance involves setting the sum of all torques about any point to zero, a necessary condition in static equilibrium.

Center of Gravity

The center of gravity is the point at which the weight of an object is considered to be concentrated. Understanding its location is vital for analyzing how weight is distributed in a system, which affects the load supported by various supports or wheels.

Load Distribution

This concept involves understanding how the total weight and any additional loads are shared among the different support points of a structure or vehicle. The relative positions of loads with respect to supports and the center of gravity determine the magnitude of the forces supported by each point.

Transcript

00:01 For this problem, we're considering the situation of two people sitting in a car.

00:06 We're told their weights, the weight of the car, as well as the location of the center of gravity of the car.

00:13 And we want to know what are the forces that are on the front and rear wheels of the car.

00:22 So this is an equilibrium equation.

00:25 The car and the people in it are not moving.

00:27 So they are in equilibrium.

00:28 So that means that we are going to apply the equilibrium conditions that the sum of the forces must be equal to zero and the sum of the torques must be equal to zero.

00:42 But in order to apply those conditions, at least for me, it's helpful to draw a more simplified diagram or a free body diagram of the situation.

00:55 So instead of drawing the car, we're just going to have a straight line here.

01:01 And i want to add in where the different forces are acting on this line.

01:07 So we're going to have the support force from the front wheels.

01:11 I'm going to call that s sub f at the front and the support force from the rear wheels, which i will call s sub r at the back.

01:22 And now we have three downward forces first.

01:26 The force downward of the person sitting in the front seat, which i've called person one, so weight one, and that acts at a distance l over 3 from the front wheel.

01:44 We then have the weight of the car itself.

01:48 The center of gravity is at 0 .4l, so the weight of the car is also going to act at that location.

02:00 So this distance here between the front and where the weight of the car is acting is 0 .4l.

02:11 And then lastly, we have the person sitting in the back seat, which i've called person 2.

02:18 And the distance here is 1 3rd l plus 1⁄2l.

02:25 So doing a little bit of adding of fractions, we see that this is a total distance of 5 .6.

02:34 6l.

02:35 Okay.

02:37 Now that we have our simplified diagram, it's pretty easy to apply our two conditions, starting with the first one.

02:43 So the sum of the forces must be equal to zero.

02:48 And we can just add them all up...

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