A 4.6-kg block slides down an inclined plane that makes an angle of 26° with the horizontal. Starting from rest, the block slides a distance of 2.6 m in 4.9 s. Find the coefficient of kinetic friction between the block and plane.
find the acceleration down the plane
distance=1/2 a t^2 solve for a.
Then, summing forces on the plane
forcedown-friction=ma
mg*sinTheta-mg*cosTheta*mu=ma
solve for mu.
To find the coefficient of kinetic friction between the block and 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:
F_gravity = m * g
where m is the mass of the block and g is the acceleration due to gravity (9.8 m/s^2).
Given: Mass of the block, m = 4.6 kg
Gravitational acceleration, g = 9.8 m/s^2
Therefore, F_gravity = 4.6 kg * 9.8 m/s^2 = 44.88 N
Step 2: Calculate the net force acting on the block.
The net force can be calculated using the formula:
F_net = F_applied - F_friction
where F_applied is the force applied on the block and F_friction is the force of friction acting on the block.
In this case, the block is sliding down the inclined plane, so the applied force is equal to the component of gravitational force acting along the incline. Thus, F_applied = F_gravity * sin(θ), where θ is the angle of the inclined plane.
Given: Angle of the inclined plane, θ = 26°
Therefore, F_applied = 44.88 N * sin(26°) = 19.26 N
Step 3: Calculate the acceleration of the block.
The acceleration of the block can be calculated using the formula:
a = (F_net) / m
where F_net is the net force acting on the block and m is the mass of the block.
Therefore, a = (19.26 N) / (4.6 kg) = 4.1957 m/s^2
Step 4: Calculate the coefficient of kinetic friction.
The coefficient of kinetic friction can be calculated using the formula:
μ_k = (F_friction) / (F_normal)
where F_friction is the force of friction acting on the block and F_normal is the normal force acting on the block.
On an inclined plane, the normal force can be calculated using the formula:
F_normal = m * g * cos(θ)
Given: Angle of the inclined plane, θ = 26°
Therefore, F_normal = 4.6 kg * 9.8 m/s^2 * cos(26°) = 39.982 N
Now, to find the force of friction, we can use the formula:
F_friction = m * a
Therefore, F_friction = 4.6 kg * 4.1957 m/s^2 = 19.26 N
Finally, we can calculate the coefficient of kinetic friction:
μ_k = 19.26 N / 39.982 N = 0.482
To find the coefficient of kinetic friction between the block and the inclined plane, we can use Newton's second law and the equations of motion.
Step 1: Determine the forces acting on the block.
The forces acting on the block are the force of gravity (mg), the normal force (N), and the force of friction (f). Since the block is sliding down the inclined plane, the force of friction acts opposite to the direction of motion.
Step 2: Resolve the forces into components.
We need to resolve the force of gravity and the normal force into components that are parallel and perpendicular to the inclined plane. The parallel component is mg * sin(θ), where θ is the angle of the inclined plane, and the perpendicular component is mg * cos(θ).
Step 3: Write the equations of motion.
Using Newton's second law, we can write the equation of motion in the direction parallel to the inclined plane:
m * a = mg * sin(θ) - f
where m is the mass of the block, a is the acceleration, and f is the force of friction.
Step 4: Find acceleration.
To find the acceleration, we can use the equation of motion for uniform accelerated motion:
s = ut + (1/2) * a * t^2
where s is the distance traveled, u is the initial velocity (which is zero in this case), a is the acceleration, and t is the time taken. Rearranging the equation, we get:
a = (2s) / (t^2)
Step 5: Calculate the coefficient of kinetic friction.
Substituting the known values in the equations, we can find the acceleration. Then, we can rearrange the equation of motion to solve for the force of friction:
f = mg * sin(θ) - m * a
Finally, we can calculate the coefficient of kinetic friction using the equation:
μ = f / (mg * cos(θ))
Plugging in the given values:
m = 4.6 kg
θ = 26°
s = 2.6 m
t = 4.9 s
g = 9.8 m/s^2 (acceleration due to gravity)
Step 4: Find acceleration.
a = (2s) / (t^2) = (2 * 2.6) / (4.9^2) = 0.225 m/s^2
Step 5: Calculate the coefficient of kinetic friction.
f = mg * sin(θ) - m * a = (4.6 * 9.8 * sin(26°)) - (4.6 * 0.225) = 40.87 - 1.04 = 39.83 N
μ = f / (mg * cos(θ)) = 39.83 / (4.6 * 9.8 * cos(26°)) = 0.48
Therefore, the coefficient of kinetic friction between the block and the plane is approximately 0.48.