A 5.4 kg block slides down an inclined plane
that makes an angle of 38� with the horizontal.
Starting from rest, the block slides a distance
of 2.5 m in 5.3 s.
The acceleration of gravity is 9.81 m/s2 .
Find the coefficient of kinetic friction be-
tween the block and plane.
Oh, let me calculate that for you, but first let me slip into my thinking clown shoes. 😄
Alright, let's get started. We need to find the coefficient of kinetic friction between the block and the plane. To do that, we'll use the following formula:
F_net = m * a
Where F_net is the net force, m is the mass of the block, and a is the acceleration. In this case, the net force can be expressed as the sum of the gravitational force acting on the block and the force of kinetic friction:
F_net = m * g * sin(theta) - m * g * cos(theta) * mu_k
Where g is the acceleration due to gravity, theta is the angle of the inclined plane, and mu_k is the coefficient of kinetic friction.
We also know that the distance traveled (s) can be calculated using the equation:
s = u * t + (1/2) * a * t^2
Where u is the initial velocity (which is zero in this case), t is the time, and a is the acceleration.
Let's find the acceleration first. Using the equation above, we can rearrange it to solve for a:
a = 2 * (s - u * t) / t^2
Substituting the given values, we have:
a = 2 * (2.5 - 0 * 5.3) / (5.3)^2
Calculating that, we find a = 0.131 m/s^2.
Now we can substitute all the values into the first equation to find the coefficient of kinetic friction:
m * a = m * g * sin(theta) - m * g * cos(theta) * mu_k
Simplifying, we have:
a = g * (sin(theta) - cos(theta) * mu_k)
Rearranging, we can solve for mu_k:
mu_k = (g * sin(theta) - a) / (g * cos(theta))
Substituting the given values, we have:
mu_k = (9.81 * sin(38) - 0.131) / (9.81 * cos(38))
Calculating that, we find mu_k ≈ 0.288.
So, the coefficient of kinetic friction between the block and the inclined plane is approximately 0.288.
Remember, sometimes math can be a bit slippery, so it's always good to double-check your calculations. 😄
To find the coefficient of kinetic friction between the block and the plane, we can use the following steps:
Step 1: Calculate the gravitational force acting on the block:
The gravitational force can be calculated using the formula:
Weight (W) = mass (m) × acceleration due to gravity (g)
W = 5.4 kg × 9.81 m/s^2
W = 52.914 N
Step 2: Calculate the net force acting on the block:
The net force can be calculated using the equation:
Net force (F_net) = mass (m) × acceleration (a)
Since the block starts from rest and slides down the inclined plane, we know that the acceleration is non-zero. Let's denote it as "a".
F_net = 5.4 kg × a
Step 3: Determine the perpendicular and parallel components of the gravitational force:
The perpendicular component (W_perpendicular) is equal to W × cos(angle of the inclined plane), and the parallel component (W_parallel) is equal to W × sin(angle of the inclined plane).
W_perpendicular = W × cos(38°)
W_parallel = W × sin(38°)
Step 4: Calculate the frictional force:
The frictional force (F_friction) can be calculated using the equation:
F_friction = coefficient of kinetic friction (μ_k) × normal force (N)
The normal force (N) can be calculated as N = W_perpendicular.
F_friction = μ_k × N
Step 5: Determine the acceleration:
Since the block slides down the inclined plane, the net force acting on the block is the difference between the parallel component of the gravitational force and the frictional force.
F_net = W_parallel - F_friction
5.4 kg × a = W_parallel - F_friction
Step 6: Solve for the acceleration:
5.4 kg × a = W_parallel - (μ_k × N)
5.4 kg × a = W_parallel - (μ_k × W_perpendicular)
Step 7: Calculate the acceleration:
Using the known values, substitute them into the equation and solve for the acceleration (a).
Step 8: Calculate the coefficient of kinetic friction:
Once we have the acceleration, we can solve for the coefficient of kinetic friction (μ_k) using the equation:
μ_k = (W_parallel - 5.4 kg × a) / W_perpendicular
By following these steps and substituting the given values, you should be able to find the coefficient of kinetic friction between the block and the plane.
To find the coefficient of kinetic friction between the block and the plane, we can use the following steps:
Step 1: Determine the net force acting on the block in the direction of motion.
The net force (Fnet) can be calculated using Newton's second law of motion, which states that the net force on an object is equal to the product of its mass (m) and acceleration (a). In this case, the acceleration of the block can be determined by the distance traveled (d) and the time taken (t) using the formula:
a = 2 * (d/t^2)
Step 2: Determine the normal force acting on the block.
The normal force (N) is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force is equal to the component of the weight of the block that acts perpendicular to the inclined plane. It can be calculated using the formula:
N = m * g * cos(θ)
where m is the mass of the block, g is the acceleration due to gravity, and θ is the angle of the inclined plane with the horizontal.
Step 3: Determine the frictional force acting on the block.
The frictional force (Ffriction) can be calculated using the coefficient of kinetic friction (μk) and the normal force (N) using the formula:
Ffriction = μk * N
Step 4: Equate the net force and the net force in the direction of motion.
Since the block is sliding down the inclined plane, the net force in the direction of motion is equal to the difference between the gravitational force component parallel to the plane (m * g * sin(θ)) and the frictional force (Ffriction). We can equate this to the net force determined in Step 1:
Fnet = m * a = m * g * sin(θ) - Ffriction
Step 5: Solve for the coefficient of kinetic friction (μk).
Rearrange the equation from Step 4 and plug in the values to solve for μk:
μk = (m * g * sin(θ) - m * a) / N
Substituting the given values:
m = 5.4 kg
g = 9.81 m/s^2
θ = 38 degrees
d = 2.5 m
t = 5.3 s
Now, perform the calculations to determine the coefficient of kinetic friction.