A mass m = 17 kg is pulled along a horizontal floor with NO friction for a distance d =5.4 m. Then the mass is pulled up an incline that makes an angle θ = 32° with the horizontal and has a coefficient of kinetic friction μk = 0.36. The entire time the massless rope used to pull the block is pulled parallel to the incline at an angle of θ = 32° (thus on the incline it is parallel to the surface) and has a tension T =75 N.
1) How far up the incline does the block travel before coming to rest? (Measured along the incline).
To find the distance the block travels up the incline before coming to rest, we need to consider the forces acting on the block.
First, let's calculate the force of gravity acting on the block. The force of gravity is given by the equation:
F_gravity = m * g
where m is the mass of the block and g is the acceleration due to gravity (which is approximately 9.8 m/s^2).
Given that the mass of the block is 17 kg, we can calculate the force of gravity:
F_gravity = 17 kg * 9.8 m/s^2 = 166.6 N
Next, let's calculate the component of the force of gravity acting parallel to the incline. This parallel component of the force of gravity is given by the equation:
F_parallel = F_gravity * sin(θ)
where θ is the angle of the incline (32° in this case). Therefore:
F_parallel = 166.6 N * sin(32°) = 88.30 N
Now, since the block is being pulled up the incline with a tension of 75 N, there is a net force acting on the block in the upward direction. The net force can be calculated by subtracting the force of friction from the tension:
Net force = T - F_friction
The force of friction can be calculated using the equation:
F_friction = μk * F_normal
where μk is the coefficient of kinetic friction, and F_normal is the normal force. Since the block is on an inclined plane, the normal force is equal to the component of the force of gravity perpendicular to the incline. Therefore:
F_normal = F_gravity * cos(θ)
F_normal = 166.6 N * cos(32°) = 141.45 N
Substituting the values into the equation for the force of friction:
F_friction = 0.36 * 141.45 N = 50.92 N
Finally, we can calculate the net force:
Net force = 75 N - 50.92 N = 24.08 N
The net force is responsible for accelerating the block up the incline. We can use Newton's second law to find the acceleration:
Net force = mass * acceleration
24.08 N = 17 kg * acceleration
Solving for acceleration:
acceleration = 1.415 m/s^2
Now, we can find the distance traveled using the equations of linear motion. Since the block starts from rest, we can use the equation:
v^2 = u^2 + 2as
where v is the final velocity (which is zero in this case), u is the initial velocity (also zero), a is the acceleration, and s is the distance traveled.
Rearranging the equation, we get:
s = -u^2 / (2 * a)
Substituting the values:
s = -(0^2) / (2 * 1.415 m/s^2)
s = 0 meters
Therefore, the block does not travel any distance up the incline before coming to rest.