A block with initial velocity of 3 m/s slides 9m across a rough horizontal surface before coming to rest. What is the coefficient of kinetic friction?
V^2=Vo^2+2ax
0=9+18x
a=-1/2
Ffr=mukFn
Sigma F=ma
ma=mukFn
To find the coefficient of kinetic friction, we need to use the following equations:
1. V^2 = Vo^2 + 2ax - This equation relates the final velocity (V), initial velocity (Vo), acceleration (a), and displacement (x).
2. Sigma F = ma - This equation represents Newton's second law, where the sum of the forces (Sigma F) acting on an object is equal to the mass (m) of the object multiplied by its acceleration (a).
3. Ffr = mukFn - This equation relates the force of friction (Ffr), coefficient of kinetic friction (muk), and the normal force (Fn).
In this case, we are assuming that the initial velocity (Vo) is positive, the final velocity (V) is zero (as the block comes to rest), and the displacement (x) is 9 meters.
Using equation 1, we can find the acceleration (a):
0 = Vo^2 + 2ax
0 = (3 m/s)^2 + 2a(9 m)
0 = 9 + 18a
By isolating "a" in the equation, we get:
-9 = 18a
a = -9/18
a = -1/2 m/s^2
Now that we have the acceleration, we can use equation 2 to find the force of friction (Ffr):
Sigma F = ma
Ffr = ma
Ffr = (-1/2 m/s^2)(m)
Looking at equation 3, we also need to find the normal force (Fn). The normal force is equal to the weight of the block, which can be calculated as:
Fn = mg
Since the block is on a horizontal surface, the normal force (Fn) must be equal and opposite to the gravitational force (mg). Therefore, Fn = mg.
Now, substituting the value of Fn into equation 3, we get:
Ffr = mukFn
Ffr = mukmg
Comparing equations 2 and 3, we can equate them as:
ma = mukmg
Canceling out the mass (m) from both sides of the equation, we get:
a = ukg
Now, we can finally find the coefficient of kinetic friction (uk) by substituting the acceleration (a = -1/2 m/s^2) and the acceleration due to gravity (g = 9.8 m/s^2):
-1/2 = uk(9.8)
uk = -1/2 divided by 9.8
uk = -0.051
Therefore, the coefficient of kinetic friction is approximately -0.051. Note that the negative sign indicates that the direction of the force of friction opposes the motion of the block.