What Is the Minimum Force Needed to Push a Tall Dresser Across the Floor?

[amcavoy]

Say you have a dresser that is to be pushed across a floor. The height is x, the width is y, and the coefficient of static friction is u. If you apply a force F and the top left corner (height x), what is the minimum value of F so the the dresser will move across the floor?

I assumed here that the center of gravity is in the middle of the dresser somewhere. My approach to this problem would be then to calculate the torque from the force F and the torque from the torque from the friction. However, I don't really know how to factor in the width of the dresser. Does anyone have any ideas on this?

Thank you.

 

[LeonhardEuler]

Well, suppose the force you exert is perfectly horizontal. Then there are four forces acting on the dresser that we are concerned with: Your pushing force, the friction force, gravity and the normal force. Balancing forces in the vertical direction tells you that the normal force is equal in magnitude to the force of gravity, mg. Now, to consider when the dresser either starts sliding or tipping over, assume the force of friction is at a maximum, $F_f=\mu N=\mu mg$. Balancing forces in the horizontal direction will tell you that the force you must exert is $F_{push}=\mu mg$. But this does not take into account the torques. When you start pushing an object the pushing force and friction force form an action reaction pair that generates a torque. As the object shifts slightly, one side of the object is pressed to the ground more forcefully than the other. This recenters the normal force. This causes the gravitational and normal forces to create a counter-torque. The object will tip over once the normal force needs to be centered ouside of the object balance the pushing and friction torque.

 

[quicknote]

I would approach this problem by first understanding the concept of static friction. Static friction is a force that opposes the motion of an object when it is at rest. In this case, the dresser is at rest and we want to push it across the floor. The coefficient of static friction (u) is a measure of how much friction is present between two surfaces, in this case, the dresser and the floor.

To determine the minimum force (F) needed to push the dresser, we need to consider the forces acting on the object. The force F applied at the top left corner will create a torque, which is a measure of the force's ability to rotate the object around a fixed point. The dresser's center of gravity will also create a torque in the opposite direction. This torque is dependent on the dresser's width (y) as it determines the distance from the center of gravity to the point of rotation (top left corner).

To find the minimum value of F, we need to consider the point at which the forces are balanced. This means that the torque from the applied force F must be equal to the torque from the center of gravity. Mathematically, this can be represented as Fy = uFx, where Fy is the torque from the applied force and Fx is the torque from the center of gravity.

Therefore, the minimum value of F can be calculated by dividing the torque from the center of gravity by the width of the dresser (F = uFx/y). This means that the wider the dresser, the lower the minimum force needed to move it.

In conclusion, to determine the minimum force needed to push a tall object such as a dresser, we need to consider the concept of static friction and the forces acting on the object. The width of the object plays a crucial role in determining the minimum force needed, and as scientists, we must consider all factors to accurately solve this problem.