A 2.90 kg block starts from rest at the top of a 30° incline and accelerates uniformly down the incline, moving 2.23 m in 1.90 s.
(a) Find the magnitude of the acceleration of the block.
(b) Find the coefficient of kinetic friction between the block and the incline.
(c) Find the magnitude of the frictional force acting on the block.
(d) Find the speed of the block after it has slid a distance 2.23 m.
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To solve this problem, we will use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration (F = m*a).
(a) To find the magnitude of the acceleration of the block, we will use the equation of motion for objects on an inclined plane. The equation is given by:
acceleration = g * sin(theta)
where g is the acceleration due to gravity (9.8 m/s^2) and theta is the angle of the incline (30°).
Substituting the values into the equation:
acceleration = 9.8 m/s^2 * sin(30°)
Calculating this expression will give us the magnitude of the acceleration.
(b) To find the coefficient of kinetic friction between the block and the incline, we can use the equation:
frictional force = coefficient of friction * normal force
However, since the block is moving down the incline and accelerating, the frictional force is acting in the opposite direction of motion and is given by:
frictional force = mass * acceleration
Therefore, we can rearrange the equation to find the coefficient of kinetic friction:
coefficient of friction = frictional force / normal force
(c) To find the magnitude of the frictional force acting on the block, we can use the equation:
frictional force = mass * acceleration
We already know the mass of the block from the problem statement (2.90 kg), and we found the acceleration in part (a). Substituting these values will give us the magnitude of the frictional force.
(d) To find the speed of the block after it has slid a distance of 2.23 m, we can use the equations of motion for objects in linear motion with constant acceleration. The equation is given by:
final velocity^2 = initial velocity^2 + 2 * acceleration * distance
Since the block starts from rest (initial velocity = 0 m/s), we can simplify the equation to:
final velocity^2 = 2 * acceleration * distance
Substituting the known values (acceleration from part (a) and distance from the problem statement) will give us the square of the final velocity. We can then take the square root to find the final velocity.