A 5.1 kg block slides down an inclined plane that makes an angle of 26 degrees with the horizontal.
Starting from rest, the block slides a distance of 2.9 m in 4.6 s.
The acceleration of gravity is 9.81 m/s2 .
Find the coefficient of kinetic friction between the block and plane.
I don't see how this can be done without knowing the velocity at the end of the distance.
0.01396
To find the coefficient of kinetic friction between the block and the plane, we can use the following steps:
Step 1: Find the acceleration of the block.
Since we know the distance traveled (2.9 m) and the time taken (4.6 s), we can use the formula of motion:
\[
s = ut + \frac{1}{2}at^2
\]
Where:
s = distance traveled
u = initial velocity (which is 0 in this case)
a = acceleration
t = time taken
Rearranging the formula, we get:
\[
a = \frac{2(s - ut)}{t^2}
\]
Plugging in the given values:
\[
a = \frac{2(2.9 - 0)}{(4.6)^2}
\]
Step 2: Resolve the weight of the block into components.
The weight of the block can be resolved into two components: the component parallel to the plane and the component perpendicular to the plane.
The component parallel to the plane (F//) is given by:
\[
F// = mg \cdot \sin(\theta)
\]
Where:
m = mass of the block = 5.1 kg
g = acceleration due to gravity = 9.81 m/s²
θ = angle made by the inclined plane with the horizontal = 26 degrees
The component perpendicular to the plane (F⊥) is given by:
\[
F⊥ = mg \cdot \cos(\theta)
\]
Step 3: Determine the net force acting on the block.
The net force acting on the block is given by:
\[
F_{net} = F_{applied} - F_{friction}
\]
Since the block is sliding down the inclined plane, the applied force is zero (F_applied = 0). Therefore, the net force acting on the block is equal to the force of friction (F_friction).
Step 4: Find the force of friction.
The force of friction is given by:
\[
F_{friction} = \mu \cdot F⊥
\]
Where:
μ = coefficient of kinetic friction (to be determined)
F⊥ = component perpendicular to the plane
Step 5: Equate the force of friction to the net force and solve for the coefficient of kinetic friction.
Since F_friction = F_net, we can equate the two equations:
\[
\mu \cdot F⊥ = F_{net}
\]
Plugging in the values of F⊥ (mg · cos(θ)) and F_net (ma), we get:
\[
\mu \cdot mg \cdot \cos(\theta) = m \cdot a
\]
Simplifying the equation:
\[
\mu = \frac{a}{g \cdot \cos(\theta)}
\]
Now we can substitute the values for a, g, and θ, and calculate the coefficient of kinetic friction (μ).