- #1
NikolasLund
- 7
- 1
- Homework Statement
- A squash ball is dropped vertically into a body of water. The balls mass is m=22,4 g and the buoyancy acting on it is ##F_{buoyancy} = 0,338 N##. The attached file shows the ball's velocity after time t.
Find the magnitude and direction of the frictional force at time t = 0,070 s.
- Relevant Equations
- Energetic approach: ##A_{ext} = \Delta E_{mech} = F_{ext}*s##
Dynamic approach: ##F_{res}=ma##; ##F_{res} = F_{grav} - F_{buoyancy} - F_{friction}##
I first attempted to solve the problem by ##A_{ext} = \Delta E_{mech} = F_{ext}s##. Here, ##F_{ext} = F_{friction}## and ##\Delta E_{mech} = E_{k2} + E_{p2} - E_{k1}##. We obtain then the following equation: $$F_{friction} = (m((v_2)^2 - (v_1)^2))/2h + mg,$$where ##v_1## is the velocity at time ##t = 0 s##, ##v_2## is the velocity at time ##t = 0,070 s## and ##h = s##.
Now, we only have to find ##h## which I did by plotting the dataset into a program (I use Logger Pro), where I choose a fitting model/function to describe the set. Then, I let the program integrate the function from ##t_0 = 0s## to ##t = 0,070s##, upon which we obtain ##h = 0,1145 m##.
The following end result is then found: $$F_{friction} = -0,41 N$$.
Now, using the Dynamic method, I get a different answer for the frictional force, and I struggle to identify what potential differences in assumption there are between the two approaches. Using this method, the resultant force on the ball at time ##t = 0,070 s## can be found: $$F_{res} = ma_{t = 0,070 s}$$. To this end, we must find the acceleration at the given time, which I did by letting Logger Pro differentiate the function, which yields ##a_{t = 0,070 s} = -14,389 m/s^2##. The resulting force is then calculated to be ##F_{res} = -0,35 N##.
The resulting force can also be expressed by $$F_{res} = F_{grav} - F_{buoyancy} - F_{friction}$$. We can calculate the gravitational force, and know the buoyancy, so we get: $$F_{friction} = 0,25 N$$
How is the difference in results to be explained? It seems to me that both approaches are correct. Also, does the negative frictional force in found in the second method not contradict the assumption that the ball is continuously sinking deeper, until it eventually reaches the bottom. Or is this assumption wrong?
Now, we only have to find ##h## which I did by plotting the dataset into a program (I use Logger Pro), where I choose a fitting model/function to describe the set. Then, I let the program integrate the function from ##t_0 = 0s## to ##t = 0,070s##, upon which we obtain ##h = 0,1145 m##.
The following end result is then found: $$F_{friction} = -0,41 N$$.
Now, using the Dynamic method, I get a different answer for the frictional force, and I struggle to identify what potential differences in assumption there are between the two approaches. Using this method, the resultant force on the ball at time ##t = 0,070 s## can be found: $$F_{res} = ma_{t = 0,070 s}$$. To this end, we must find the acceleration at the given time, which I did by letting Logger Pro differentiate the function, which yields ##a_{t = 0,070 s} = -14,389 m/s^2##. The resulting force is then calculated to be ##F_{res} = -0,35 N##.
The resulting force can also be expressed by $$F_{res} = F_{grav} - F_{buoyancy} - F_{friction}$$. We can calculate the gravitational force, and know the buoyancy, so we get: $$F_{friction} = 0,25 N$$
How is the difference in results to be explained? It seems to me that both approaches are correct. Also, does the negative frictional force in found in the second method not contradict the assumption that the ball is continuously sinking deeper, until it eventually reaches the bottom. Or is this assumption wrong?
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