A 9.0-kg box of oranges slides from rest down a frictionless incline from a height of 5.0 m. A constant frictional force, introduced at point A, brings the block to rest at point B, 19 m to the right of point A.
What is the coefficient of kinetic friction, k , of the surface from A to B?
0.33
0.11
0.26
0.52
0.47
0.26
To find the coefficient of kinetic friction, we need to use the given information about the box of oranges sliding down the incline.
First, we need to calculate the work done by the frictional force from point A to point B. The work done by friction is equal to the force of friction multiplied by the distance along which it acts. In this case, the force of friction is acting opposite to the direction of motion, so it is negative.
The work done by friction is given by the formula:
Work = -Force of friction * Distance
Next, we need to find the work done by gravity as the box slides down the incline. The work done by gravity is equal to the weight of the box multiplied by the vertical distance it moves.
The work done by gravity is given by the formula:
Work = Weight * Vertical distance
The weight of the box is equal to its mass multiplied by the acceleration due to gravity.
Weight = Mass * Acceleration due to gravity
Now, since the incline is frictionless, the work done by gravity should be the same as the work done by friction. Therefore, we can equate these two expressions for work:
-Force of friction * Distance = Weight * Vertical distance
Simplifying this equation, we get:
-Force of friction = (Mass * Acceleration due to gravity * Vertical distance) / Distance
We are given the mass of the box (9.0 kg), the vertical distance (5.0 m), and the distance from A to B (19 m). We also know that the acceleration due to gravity is approximately 9.8 m/s^2.
Plug these values into the equation to find the force of friction:
-Force of friction = (9.0 kg * 9.8 m/s^2 * 5.0 m) / 19 m
Calculating this expression, we find:
-Force of friction ≈ -22.947 kg * m/s^2
Finally, we can find the coefficient of kinetic friction (k) using the equation:
Force of friction = k * Normal force
In this case, the normal force is equal to the weight of the box since there is no vertical acceleration. So we can write:
-Force of friction ≈ k * Weight
Since we already found the force of friction to be approximately -22.947 kg * m/s^2, we can rearrange this equation to solve for the coefficient of kinetic friction (k):
k = (-Force of friction) / Weight
Substituting the values, we get:
k ≈ (-(-22.947 kg * m/s^2)) / (9.0 kg * 9.8 m/s^2)
Calculating this expression, we find:
k ≈ 0.26
Therefore, the coefficient of kinetic friction, k, of the surface from A to B is approximately 0.26.
To find the coefficient of kinetic friction, we can use the work-energy principle.
1. The potential energy of the box at point A is given by PE = mgh, where m is the mass of the box (9.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (5.0 m).
PE = (9.0 kg)(9.8 m/s^2)(5.0 m) = 441 J
2. The work done by the frictional force is given by W_f = -Fd, where F is the frictional force and d is the distance travelled (19.0 m to the right). Since the box comes to rest, the work done by the frictional force is equal to the change in potential energy.
W_f = -Fd = -md^2/2
-md^2/2 = 441 J
3. From the given options, we can use the given distances to find the value of d^2:
d = 19 m, d^2 = 19^2 = 361
4. We can then solve for the coefficient of kinetic friction using the equation:
-md^2/2 = 441 J
-(9.0 kg)(361/2) = 441 J
-1624.5 kg*m^2/s^2 = 441 J
5. Rearranging the equation gives us:
1624.5 kg*m^2/s^2 = -441 J
6. Dividing both sides of the equation by -441 J gives us:
1624.5 kg*m^2/s^2 / -441 J = 1
7. Finally, we can take the square root of both sides to solve for the coefficient of kinetic friction:
sqrt(1624.5 kg*m^2/s^2 / -441 J) = sqrt(1)
sqrt(3.682) ≈ 1
Based on the given options, the coefficient of kinetic friction, k, is approximately 0.33 (option 1).