Max Accel of Object: Can't Find Solution?

[thepatient]

Homework Statement

This problem seems so simple yet I don't feel there is enough information to know what is the maximum acceleration the object can travel at without tipping over. I uploaded a picture of the object along with the variables.

rope AB = 6 ft

Angle of AB = 78 degrees

Acceleration positive to the right.

Homework Equations

∑F = m*a

∑M = I*α + r*m*a, where r is the distance between the center of mass and location at which the moments are being taken, and a is the acceleration at the center of mass of the object.

The Attempt at a Solution

Drawing the free body diagram of the entire cart, there is no normal force acting on the rear wheel, since this is the limiting condition for tipping over.

I tried taking the cart apart at the point where the horizontal board meets the vertical board. I set reaction forces at the joint, a weight of m1g at the center of the board and the tension force Tab, along with unknown acceleration a.

I was able to figure going this way that the minimum tension of the rope, using the sum of the moments at the joint, must be .5*m1g/sin(78) since:

∑ M(joint) = I*α + r*m*a

but we want to avoid tipping, so alpha is zero. And the vector r is zero as well at the reaction points, so:
∑ M(joint) = 0, ccw positive.
6*cos(78)/2 *m1g + 6*cos(78)*Tabsin(78) = 0

Tab = .5m1g/sin(78). Anything larger for Tab than this will cause the object to rotate.

That's as far as I got. I couldn't use the sum of the moments in the second body consisting of vertical board and block since there is a lack of dimensions.

Maybe this is not the right way to approach this problem? It doesn't seem to make much sense because there is no external force acting on the body to make it accelerate in the first place. I'm so stuck. :(

 

[anathema]

Thank you for sharing your thoughts and approach to this problem. I understand your frustration with the lack of information provided in the problem statement. I can assure you that this is often the case in real-world scenarios and it is our job to use our knowledge and problem-solving skills to make assumptions and come up with a reasonable solution.

Based on the information provided, I would approach this problem by first identifying the center of mass of the object and then considering the forces acting on it. Since the acceleration is positive to the right, we can assume that there is a net force acting on the object in that direction. This force can be a combination of the tension force in the rope and the force of friction at the rear wheel.

To avoid tipping over, the net torque acting on the object must be zero. This means that the torque due to the tension force must be equal and opposite to the torque due to the friction force. Using the equation you provided, ∑M = I*α + r*m*a, we can calculate the maximum tension force that the rope can exert without causing the object to tip over.

However, as you mentioned, there is no external force acting on the object to make it accelerate in the first place. In this case, we can assume that the object is being pushed or pulled by an external force, which is not shown in the problem statement. This external force can be taken into consideration by adding it to the net force acting on the object.

I hope this helps you approach the problem in a different way. Remember, as scientists, we often have to make assumptions and use our problem-solving skills to come up with solutions. Keep up the good work and don't get discouraged!