An object starts sliding up a plane tilted at an angle of alpha = 5.0◦ to
the horizontal. The object’s initial speed is v = 2.0m/s. How far from
the start point does the object come to a stop if μK = 0.20?
Kietic friction will apply. There will be a backwards friction force opposing motion that is equal to
M g cos 5 * ìK = 1.95 m/s^2 * M
Motion will stop when the initial kinetic energy equals the gain in potential energy PLUS the work done against friction. Let X be the distance travelled. X sin 5 is the elevation gain.
(1/2) M V^2 = Mg cos5 ìK + MgX sin 5
Cancel out the M's and solve for X
V^2 = 2g [cos5 ìK + sin 5 X]
To find the distance that the object comes to a stop, we can first calculate the object's acceleration using the angle of the plane and the coefficient of kinetic friction. Once we have the acceleration, we can then use the kinematic equation to calculate the distance traveled.
Step 1: Calculate the acceleration:
The component of gravity acting down the inclined plane is given by:
Fg_parallel = m * g * sin(alpha)
where m is the mass of the object, g is the acceleration due to gravity, and alpha is the angle of the plane.
The frictional force opposing the motion is given by:
F_friction = μk * N
where μk is the coefficient of kinetic friction and N is the normal force. The normal force can be calculated as:
N = m * g * cos(alpha)
The net force acting up the plane is given by:
F_net = Fg_parallel - F_friction
Since F_net = m * a (Newton's second law), we can solve for acceleration:
a = (Fg_parallel - F_friction) / m
Step 2: Calculate the distance traveled:
We can use the kinematic equation to find the distance traveled by the object:
v_f^2 = v_i^2 + 2 * a * s
where v_f is the final velocity (which is 0 since the object comes to a stop), v_i is the initial velocity, a is the acceleration, and s is the distance traveled.
Solving for s, we get:
s = (v_f^2 - v_i^2) / (2 * a)
Given that v_f = 0 and v_i = 2.0 m/s, and we have previously calculated the acceleration, we can substitute these values into the equation to find the distance traveled.
Let's calculate step by step:
Step 1: Calculate the acceleration:
alpha = 5.0 degrees
μk = 0.20
Fg_parallel = m * g * sin(alpha)
F_friction = μk * N
N = m * g * cos(alpha)
F_net = Fg_parallel - F_friction
a = (Fg_parallel - F_friction) / m
Step 2: Calculate the distance traveled:
v_f = 0 m/s
v_i = 2.0 m/s
s = (v_f^2 - v_i^2) / (2 * a)
Now, substitute the known values into the equations and solve for s.