n the figure below, the coefficient of kinetic friction between the block and inclined plane is 0.18 and angle θ = 55°. What is the block's acceleration (magnitude and direction) assuming the following conditions?
(a) It is sliding down the slope.
m/s2
down the slope
(b) It has been given an upward shove and is still sliding up the slope.
m/s2
down the slope
To find the block's acceleration in each condition, we can 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_net = m * a
In both conditions, we need to consider the forces acting on the block: the gravitational force (mg) and the frictional force (friction). The direction of the acceleration will depend on the net force acting on the block.
(a) When the block is sliding down the slope:
In this case, the gravitational force is acting in the downward direction, while the frictional force opposes the motion and acts in the opposite direction of the block's velocity. Therefore, we can write:
F_net = mg - friction
To determine the frictional force, we use the formula:
friction = μ * N
where μ is the coefficient of kinetic friction and N is the normal force. The normal force is the force exerted by the inclined plane on the block, perpendicular to the surface. It can be determined by decomposing the weight of the block into its components:
N = mg * cos(θ)
Substituting this into the equation for friction, we have:
friction = μ * mg * cos(θ)
Now we can substitute the expressions for friction and N back into the equation for net force:
F_net = mg - μ * mg * cos(θ)
Now, we can solve for the acceleration by dividing both sides of the equation by the mass (m):
a = g - μ * g * cos(θ)
Given that the coefficient of kinetic friction μ = 0.18 and the angle θ = 55°, we can substitute these values into the equation:
a = 9.8 m/s² - 0.18 * 9.8 m/s² * cos(55°)
Using a calculator, we can find that the block's acceleration down the slope is approximately 5.47 m/s².
Therefore, the block's acceleration, when sliding down the slope, is 5.47 m/s² in the downward direction.
(b) When the block has been given an upward shove and is still sliding up the slope:
In this case, the gravitational force is acting in the downward direction, while the frictional force helps to slow down the block's upward motion. The net force is given by:
F_net = mg + friction
Using the same formulas as in part (a) to calculate the frictional force and substituting them into the equation for net force, we have:
F_net = mg + μ * mg * cos(θ)
Again, we can solve for the acceleration by dividing both sides of the equation by the mass (m):
a = g + μ * g * cos(θ)
Substituting the coefficient of kinetic friction μ = 0.18 and the angle θ = 55° into the equation, we have:
a = 9.8 m/s² + 0.18 * 9.8 m/s² * cos(55°)
Using a calculator, we can find that the block's acceleration when sliding up the slope is approximately 7.69 m/s².
Therefore, the block's acceleration, when it has been given an upward shove and is still sliding up the slope, is 7.69 m/s² in the downward direction.