In the figure, a block of mass m is moving along the horizontal frictionless surface
with a speed of 5.70 m/s. If the slope is 11.0° and the coefficient of kinetic friction
between the block and the incline is 0.260, how far does the block travel up the incline?
To find how far the block travels up the incline, we need to break down the forces acting on the block and apply Newton's laws of motion. Here are the steps to follow:
1. Draw a free-body diagram of the block.
2. Identify the forces acting on the block. In this case, there are three main forces:
- The weight (mg) acting vertically downward.
- The normal force (N) acting perpendicular to the incline.
- The frictional force (fk) acting parallel to the incline and opposite to the motion.
3. Resolve the weight force into components. The weight can be broken down into two components: one parallel to the incline (mg*sin(theta)) and one perpendicular to the incline (mg*cos(theta)).
4. Determine the normal force (N). The normal force equals the perpendicular component of the weight force (N = mg*cos(theta)).
5. Calculate the frictional force (fk). The frictional force can be found by multiplying the coefficient of kinetic friction (μk) by the normal force (fk = μk * N).
6. Use Newton's second law (Fnet = ma) to find the net force acting on the block in the x-direction. The net force is the difference between the parallel component of the weight force and the frictional force (Fnet = mg*sin(theta) - fk).
7. Apply the kinematic equation (vf^2 = vi^2 + 2ax) to find the acceleration (a). The initial velocity (vi) is given as 5.70 m/s, and the final velocity (vf) is 0 m/s (block comes to rest). Rearranging the equation, we have a = -vi^2 / (2x), where x is the distance traveled up the incline.
8. Plug in the values and solve for the distance traveled, x.
Do you want to proceed to step 1?
To find how far the block travels up the incline, we can use the concept of work and energy. The work-energy principle states that the work done on an object is equal to the change in its kinetic energy.
Step 1: Determine the work done by the applied force:
Since the surface is frictionless, the only force doing work on the block is the component of the gravitational force that is parallel to the incline. This force can be calculated using the equation:
F_parallel = m * g * sin(theta)
where,
m = mass of the block
g = acceleration due to gravity (approximately 9.8 m/s^2)
theta = angle of the slope (11.0°)
Step 2: Calculate the work done by the friction force:
The friction force opposes the motion of the block, so it does negative work on the block. The magnitude of the friction force can be calculated using the equation:
F_friction = m * g * cos(theta) * mu
where,
mu = coefficient of kinetic friction (0.260 in this case)
The work done by the friction force can be calculated by multiplying the magnitude of the friction force by the displacement of the block along the incline.
Step 3: Calculate the change in the kinetic energy:
The change in the kinetic energy of the block is equal to the work done by the applied force minus the work done by the friction force.
Step 4: Use the work-energy principle:
Using the work-energy principle, we equate the change in kinetic energy to the work done by the applied force minus the work done by the friction force.
Change in kinetic energy = Work applied - Work friction
Since the block starts at rest vertically and its initial kinetic energy is zero, the change in kinetic energy is equal to the final kinetic energy:
1/2 * m * v^2 = (m * g * sin(theta) * d) - (m * g * cos(theta) * mu * d)
Here, v is the final velocity of the block, which is also equal to 0 (since it comes to a stop at the top of the incline).
Solving the equation for d (the distance traveled up the incline), we get:
d = (v^2) / (2 * g * (sin(theta) - mu * cos(theta))
Substituting the given values into the equation will give you the distance traveled by the block up the incline.