How to Calculate Velocity of a 4-Wheeled Vehicle Using a Pulley System

[henry1117]

Ok basically i need help to start off a question as a part of a research type thing.

i need to find out the velocity of a 4 wheeled vehicle if string is wrapped around the rear axel of the vehicle multiple times and the string then extends up above the rear axel onto a pulley where a weight sits.

Basically, as the weight is let go and falls, the vehicle starts moving and i need to know the velocity of the vehicle after the weight has fallen.

so ye, if anyone could start me off on this it would be greatly appreciated

 

[CaptainEvil]

Ok basically i need help to start off a question as a part of a research type thing.

i need to find out the velocity of a 4 wheeled vehicle if string is wrapped around the rear axel of the vehicle multiple times and the string then extends up above the rear axel onto a pulley where a weight sits.

Basically, as the weight is let go and falls, the vehicle starts moving and i need to know the velocity of the vehicle after the weight has fallen.

so ye, if anyone could start me off on this it would be greatly appreciated

As in all Classic Newton problems, let's start off with Free body diagrams, indicating the forces acting on all bodies of interest. It looks like our FBD for our mass A attatched to the pulley is only acted upon by gravity, downwards, in what we'll call the positive y direction, and tension upwards from the pulley, which we'll call T₁ (Lets assume no friction in the pulley)
Our FBD for the car shows a positive tension in our x axis, called T₂, and a negative friction along the table (or road) working against the direction of motion. There is also the force of gravity working down on the car, and a normal force acting upwards, but since the table is rigid, and the car is not falling through the road, we can show that F_gcar = F_N

Now we will use Newton's 2nd law to derive two important equations that will lead to solving for our unknowns.

Looking at the mass on the pulley, the force in the y direction will be the sum of the forces acting on it, which is simply gravity, and that will equal ma

F_blocky = F_g - T₁ = M_blocka_block

Next let's look at the motion of the car in the x direction:

F_carx = T₂ - F_f = M_cara_car

Now that we have these two equations, its time to look at some constraints.

We've already discussed that F_gcar = F_N for a rigid body, so M_carg = N (our normal force, which we will need to solve for the force of friction)

Since the whole system is moving together, we can assume that a_block = a_car = a

To find our frictional force F_f, it is given by \muN, where \mu is the coefficient of friction.

So, with these two equations, as long as you know 3 of the following unknowns: T₁, T₂, M_block, M_car, \mu, you can use your two equations and two unknowns to solve for acceleration a. Once you know that, you can describe the speed of your system (and hence your car) at any given time at any position, given simple kinematic equations that can be looked up.

hope this helps,

cheers