A 16kg sled starts up a 28 degree incline with a speed of 2.4m/s . The coefficient of kinetic friction is = 0.27.
part a)How far up the incline does the sled travel?
part b)What condition must you put on the coefficient of static friction if the sled is not to get stuck at the point determined in part (a)?
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To solve this problem, we will use the principles of physics, specifically Newton's laws of motion and the work-energy theorem.
Part a) To find the distance traveled by the sled up the incline, we need to determine the net force acting on the sled in the direction of motion. We can start by calculating the gravitational force acting on the sled. The gravitational force is given by the formula:
F_gravity = mass * acceleration due to gravity
F_gravity = 16kg * 9.8m/s^2 = 156.8N
Next, we need to determine the force of friction acting on the sled. The force of friction is equal to the coefficient of kinetic friction multiplied by the normal force. The normal force is the force exerted perpendicular to the incline and is equal to the gravitational force, given that the sled is not accelerating perpendicular to the incline. Therefore:
F_friction = coefficient of kinetic friction * normal force
F_friction = 0.27 * F_gravity = 0.27 * 156.8N = 42.456N
The net force acting on the sled is the difference between the applied force and the force of friction. In this case, the applied force is zero, as the sled is sliding up the incline due to the initial speed:
Net force = 0 - F_friction
Net force = -42.456N
Next, we need to calculate the acceleration of the sled. To do this, we use Newton's second law of motion, which states that the net force is equal to the mass of the object multiplied by its acceleration:
Net force = mass * acceleration
-42.456N = 16kg * acceleration
acceleration = -42.456N / 16kg
acceleration = -2.6535m/s^2
The negative sign indicates that the acceleration is in the opposite direction of motion.
Now, we can use the work-energy theorem to find the distance traveled by the sled. The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy:
Work = change in kinetic energy
The only force doing work on the sled is the force of gravity. The work done by gravity is equal to the product of its force and the distance traveled:
Work_gravity = F_gravity * distance
Since the sled starts from rest and the speed is given as 2.4m/s, the initial kinetic energy is zero, and the final kinetic energy is given by:
Kinetic energy = (1/2) * mass * final velocity^2
Kinetic energy = (1/2) * 16kg * (2.4m/s)^2
Kinetic energy = 27.648J
Using the work-energy theorem, we get:
27.648J = 156.8N * distance
Therefore, the distance traveled up the incline by the sled is:
distance = 27.648J / 156.8N
distance = 0.176m (rounded to three decimal places)
Part b) To ensure that the sled does not get stuck at the point determined in part a, the force of static friction must be greater than or equal to the force of gravity acting on the sled. The force of static friction can be calculated using the same formula as the force of kinetic friction:
F_static_friction = coefficient of static friction * normal force
If the force of static friction is greater than or equal to the force of gravity, the sled will not slide back down the incline.
Therefore, the condition that must be placed on the coefficient of static friction is:
coefficient of static friction >= (force of gravity) / (normal force)
coefficient of static friction >= (156.8N) / (156.8N)
coefficient of static friction >= 1
In this case, the coefficient of static friction should be greater than or equal to 1 to prevent the sled from getting stuck at the point determined in part a.