A small block of mass 0.8kg is placed at the top of a 4m plane, when it is released, it take 3.5 seconds to slide to the bottom. Determine the coefficient of friction, stating assumptions made.
Johnny is 52 inches tall Max is 65 inches tall. What percent of Johnny's height is Max's height.
113%
110%
To determine the coefficient of friction, we need to use the equations of motion for motion on an inclined plane.
Assumptions made:
1. The plane is frictionless except for the specific region where the block is sliding.
2. The block is a point mass and there is no rolling or rotational motion involved.
3. The acceleration due to gravity is constant and there is no air resistance.
Let's break down the problem into steps:
Step 1: Calculate the acceleration of the block.
Using the formula for distance traveled down an inclined plane, we have:
d = 0.5 * a * t^2,
where d is the distance traveled (4m), a is the acceleration, and t is the time taken (3.5s).
Rearranging the equation, we have:
a = (2 * d) / (t^2)
Plugging in the values, we get:
a = (2 * 4) / (3.5^2)
a ≈ 0.326 m/s^2
Step 2: Calculate the gravitational force acting on the block.
The gravitational force is given by:
F_gravity = m * g,
where m is the mass of the block (0.8kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Plugging in the values, we get:
F_gravity = 0.8 * 9.8
F_gravity ≈ 7.84 N
Step 3: Determine the normal force.
The normal force is equal in magnitude but opposite in direction to the component of the gravitational force perpendicular to the plane. This can be calculated as:
F_normal = m * g * cos(angle),
where angle is the angle of inclination of the plane.
Since the block is sliding down the plane, the angle of inclination is the same as the angle of the plane itself. We can take the cosine of the angle as the ratio of the adjacent and hypotenuse sides of the triangle formed by the plane.
For a vertical plane, the angle would be 90 degrees and cos(90 degrees) = 0.
For an inclined plane, the angle is less than 90 degrees, so 0 < cos(angle) ≤ 1.
In this case, since the block is sliding down the plane, the angle is not given explicitly. However, it can be determined using trigonometry. Let's assume it is θ.
Plugging in the values, we have:
F_normal = 0.8 * 9.8 * cos(θ)
Step 4: Calculate the frictional force.
The frictional force can be calculated using the equation:
F_friction = μ * F_normal,
where μ is the coefficient of friction.
Since we assume the plane is frictionless except for the specific region where the block is sliding, the frictional force is equal to the gravitational force acting parallel to the plane, which is given by:
F_friction = m * g * sin(angle),
Plugging in the values, we have:
F_friction = 0.8 * 9.8 * sin(θ).
Step 5: Determine the coefficient of friction.
Since we know that the frictional force is equal to the gravitational force parallel to the plane, we can equate the two equations:
m * g * sin(θ) = μ * m * g * cos(θ).
Simplifying the equation, we can cancel out the mass and acceleration due to gravity terms:
sin(θ) = μ * cos(θ).
Dividing both sides by cos(θ), we get:
tan(θ) = μ.
To determine the coefficient of friction, we need to find the value of tan(θ). We can do this using the values we have:
θ = sin^(-1)(a / g),
Plugging in the values, we get:
θ ≈ sin^(-1)(0.326 / 9.8)
θ ≈ 0.033 radians.
Finally, we can calculate the coefficient of friction:
μ = tan(θ)
μ = tan(0.033)
Using a calculator, we find that μ ≈ 0.033. Therefore, the coefficient of friction is approximately 0.033.