How to find the frictional force acting on an object (not the friction coefficient)?

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How to find the frictional force acting on an object (not the friction coefficient)?

Physics
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Let's analyze a simple case of a flat surface with a box sliding across it due a force. The frictional force always opposes the motion. If we balance the forces, we can come up with the following expression\[\sum {\bf \vec F} = m*\vec a = \vec {\bf F} - {\bf \vec F_f}\]Therefore, to determine the frictional force, without knowledge of the coefficient of friction, we need to know the mass, the acceleration, and the applied force. This, of course, is for the kinematic force of friction. To determine to the force of static friction, the box will not move, therefore we get the following expression\[\sum {\bf \vec F} = 0 = {\bf \vec F - \vec F_f}\]In this case, we must known the applied force.
|dw:1327218114124:dw| The ball's mass is 3.5 kg, the spike is 0.8 kg and the ball starts at 1.6 m above ground. I found the speed of the ball upon striking the spike to be 5.6 m/s. After collision speed is 4.56 m/s Initial Kinetic energy is 54.9 J Final Kinetic energy is 44.7 J My trying to solve the following question: "Aws a result of the ball striking the spike, the spike is driven (7.3)(10^-2) meters into the wall. Calculate the constant friction force F between the spike and the ball."
There are a couple of ways to solve this problem. I'll use the Work-Energy Theorem. First, I'll define a couple of key quantities. \[W = \Delta E\]\[W = \int\limits {\bf \vec F} ~ dx\]The only force acting on the spike is the force of friction. Therefore, \[{\bf \vec F_f}*x = \Delta KE\]From the problem statement, we know \(\Delta KE\) and \(x\). We can solve for \(\bf \vec F_f\)

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Im sorry, I'm very much a beginner, does the integral (I just know what it´s called, not what it means) mean "change in Force", I assume d is displacement but is x the "cos x"? I've only used the equation Work=(F)(displacement)(cos x), so I'm not sure.
When force is constant, like it will be here, the expression\[W = \int\limits {\bf \vec F} ~ dx\]simplifies to \[W={\bf \vec F} x\] where \(\bf \vec F\) is the component of the force in the direction of the displacement. If the force is at some angle from the direction of displacement, we need to cosine term.
So, 10.2/(0.073) = 136.98 ? Why isn't that the force the ball is applying on the spike? How would I calculate it?
We don't have enough information to determine the force the ball exerts on the spike. We need the time the ball is in contact with the ball. With this information, \[{\bf \vec F} = {\Delta \vec p \over \Delta t}\]which is the change in momentum over the change in time. Additionally, if we knew average acceleration of the spike, we could determine the force of the spike as such\[{\bf \vec F_b - \vec F_f} = m_s *a_s\]
Just to be sure I understand, then, the 10 J lost on collision are caused by friction, and that's why the change in kinetic energy is used for calcualting friction? I know it may sound obvious butI want to make sure I understand this correctly.
Correct. Energy is always conserved. Since at the end of the impact, the spike still has zero kinetic energy, the energy must be converted to other forms (namely heat) through friction.
Thanks alot!
My pleasure.

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