Physics Free Body Diagrams

Question

A cart of heavy groceries needs to be pushed out to the car. Is it better for Ahmad to push down and forward with 455N at an angle of 25° or Zahida to push 350N horizontally. There is friction (0.3 is the coefficient of friction.) The cart has a mass of 89kg

Answer 1

Okay!!

So this problem has two components.

First of all, the question of who provides more horizontal pushing force. To answer that, we have to decompose the vectors into simple horizontal and vertical components.

Zahida's vector is already purely horizontal, so we don't have to change anything. She's providing 350 Newtons of force.

Ahmad, however, is pushing at an angle. So we can draw a triangle, where his 455 Newton pushing force creates the hypotenuse of a right triangle and one of the angles is the angle he's pushing at- 25°.

So, the vertical component (y) is the side opposite the small 25° angle, and the horizontal (x) is adjacent. That means that

sin(25)= y/455N

cos(25)=x/455N

This gives us a vertical component of 192.3N, and a horizontal component of 412.4N.

On its own, this would seem to indicate that Ahmad is pushing the cart harder. However, the second part of this problem is the friction.

On it's own, the cart, with a frictional coefficient of .3, will be exerting a force on the ground, equal to it's mass times the acceleration due to gravity. So,

89kg×9.8m/s²=872.2N

Multiply that by .3 to get the amount of the force of friction, and you get 261.6N.

So in both cases, the pusher has 261.6N of force to overcome to keep the cart moving. However, Ahmad is pushing the cart down into the ground with 192.3N of force by not pushing in a straight horizontal line. So,

192.3N×.3=57.7N

He's adding an additional 57.7N of friction because by pushing the cart downward slightly, he's effectively increasing it's weight and therefore the force due to friction.

So, Zahira, when we subtract the friction of the cart's weight, is applying a net force of...

350N−261.6N=88.3N

... 88.3N

While Ahmad, when we add in the additional friction from him pushing downward to the friction of the cart's weight, is applying a net force of...

455N-(261.6N+57.7N)=135.6N

... 135.6N. This means that, even though Ahmad is indeed making the act of pushing the cart harder for himself (and he could push it faster if he just lowered the angle he's pushing at from), he's still going to push the cart faster than Zahira.