Final answer:
A) The acceleration of the block is 1.6 m/s². B) The coefficient of kinetic friction between the block and the incline is 0.2906. C) The frictional force acting on the block is 16.99 N. D) The speed of the block after sliding 2m is 2.29 m/s.
Step-by-step explanation:
A) To find the acceleration of the block, we can use the equation:
acceleration = (final velocity - initial velocity) / time
Since the block starts from rest, the initial velocity is 0 m/s. The final velocity can be found using the equation:
final velocity = initial velocity + acceleration * time
Substituting the given values, we get:
final velocity = 0 + acceleration * 1.25
From the given information, the block slides 2 m down the incline in 1.25 s, so the final velocity is 2 m/s.
Substituting the values again, we get:
2 = 0 + acceleration * 1.25
Simplifying the equation, we find:
acceleration = 2 / 1.25
B) To find the coefficient of kinetic friction, we can use the equation:
frictional force = coefficient of friction * normal force
The normal force can be found using the equation:
normal force = mass * gravitational acceleration * cos(incline angle)
Simplifying the equation, we get:
normal force = 3.97 kg * 9.8 m/s^2 * cos(29.6 degrees)
From the given information, we know the frictional force is equal to the mass of the block multiplied by the acceleration of gravity multiplied by the sine of the angle of the incline. Substituting the values, we get:
4.86 N = 3.97 kg * 9.8 m/s^2 * sin(29.6 degrees)
From the equation for the frictional force and the normal force, we can use the equation for the coefficient of kinetic friction to find:
coefficient of kinetic friction = frictional force / normal force
Substituting the values, we get:
coefficient of kinetic friction = 4.86 N / (3.97 kg * 9.8 m/s^2 * cos(29.6 degrees))
Simplifying the equation, we find:
coefficient of kinetic friction = 4.86 N / (3.97 kg * 9.8 m/s^2 * 0.8789)
C) To find the frictional force acting on the block, we can use the equation:
frictional force = coefficient of friction * normal force
We have already calculated the coefficient of kinetic friction from part B. Substituting the values, we get:
frictional force = coefficient of kinetic friction * normal force
frictional force = (4.86 N) * (3.97 kg * 9.8 m/s^2 * cos(29.6 degrees))
Simplifying the equation, we find:
frictional force = 4.86 N * (3.97 kg * 9.8 m/s^2 * 0.8789)
D) To find the speed of the block after sliding 2 m, we can use the equation:
final velocity^2 = initial velocity^2 + 2 * acceleration * distance
The block starts from rest, so the initial velocity is 0 m/s. The distance is given as 2 m. Substituting the values, we get:
final velocity^2 = 0^2 + 2 * acceleration * 2
final velocity^2 = 4 * acceleration
Since we have already calculated the acceleration in part A, we can substitute the value:
final velocity^2 = 4 * (2 / 1.25)
Simplifying the equation, we find:
final velocity = √(4 * (2 / 1.25))