a 5 kg concrete block accelerates down a 34 degree slope at 4.2m/s^2. Find the coefficient of friction between the block and the slope.
The net force along the slope that produces the acceleration is
F = M g sin 34 - u M g cos 34 = M a
where u is the coefficient of friction.
M cancels out-- you don't need to know it. You can solve for u
I get u = 0.158, but don't trust my math. Check my thinking and do the numbers.
Student
Well, I'm not one to trust when it comes to math, but I'll certainly give it a shot! Let's break it down, step by step.
The net force along the slope is given by:
F = M * g * sin(34) - u * M * g * cos(34)
Where:
- F is the net force along the slope,
- M is the mass of the block (5 kg),
- g is the acceleration due to gravity (approximately 9.8 m/s^2),
- u is the coefficient of friction.
We also know that the acceleration of the block down the slope is given as 4.2 m/s^2.
So we have:
4.2 = 5 * 9.8 * sin(34) - u * 5 * 9.8 * cos(34)
Simplifying further, we get:
4.2 = 5 * 9.8 * 0.559 - u * 5 * 9.8 * 0.834
Now let's solve for u:
u * 5 * 9.8 * 0.834 = 5 * 9.8 * 0.559 - 4.2
u * 41.07 = 27.3042
u = 27.3042 / 41.07
u ≈ 0.665
So, according to my "clown calculations," the coefficient of friction between the block and the slope is approximately 0.665. Please double-check my math before trusting this answer!
To find the coefficient of friction, we'll start by analyzing the forces acting on the concrete block. We'll use Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration.
In this case, the net force along the slope is the force component parallel to the slope. It is given by:
F = M g sinθ - u M g cosθ = M a
Where:
M is the mass of the concrete block (5 kg)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
θ is the angle of the slope (34 degrees)
u is the coefficient of friction
a is the acceleration (4.2 m/s^2)
Now we can solve for u.
Rearranging the equation, we have:
M g sinθ - u M g cosθ = M a
Multiplying both sides by M:
M g sinθ - u M g cosθ = M a
Canceling out the mass (M):
g sinθ - u g cosθ = a
Rearranging further:
u g cosθ = g sinθ - a
Dividing both sides by g cosθ:
u = (g sinθ - a) / (g cosθ)
Now we can substitute the given values into the equation and calculate u.
u = (9.8 m/s^2 * sin(34°) - 4.2 m/s^2) / (9.8 m/s^2 * cos(34°))
Calculating the right side of the equation yields:
u = (5.288 - 4.2) / 8.022
Simplifying further:
u = 1.088 / 8.022
Finally, we can divide to find the answer:
u ≈ 0.135
Therefore, the coefficient of friction between the concrete block and the slope is approximately 0.135.