A block of mass 0.78kg starts from rest at point A and slides down a frictionless hill of height h. At the bottom of the hill it slides across a horizontal piece of track where the coefficient of kinetic friction is 0.29. This section (from points B to C) is 4.08m in length. The block then enters a frictionless loop of radius r= 2.87m. Point D is the highest point in the loop. The loop has a total height of 2r. Note that the drawing below is not to scale.
Question???????
To solve this problem, we need to find different parameters such as the height of the hill, the velocity of the block at different points, and whether it completes the loop or not.
Let's analyze the motion of the block at different stages:
1. Sliding down the hill (From Point A to Point B):
The potential energy is converted into kinetic energy as the block slides down the hill without friction.
According to the law of conservation of energy:
m * g * h = 1/2 * m * v₁²
where m is the mass of the block, g is the acceleration due to gravity, h is the height of the hill, and v₁ is the velocity of the block at point B.
Rearranging the equation, we can solve for h:
h = (v₁²) / (2 * g) ...Equation 1
2. Sliding across the horizontal track (From Point B to Point C):
The block experiences friction on the horizontal track.
The frictional force is given by the equation:
f_friction = coefficient of kinetic friction * normal force
The normal force is equal to the weight of the block, which is m * g.
The work done by friction over a displacement is equal to the force of friction multiplied by the displacement.
Therefore, the work done by friction is:
work_friction = f_friction * displacement = coefficient of kinetic friction * m * g * displacement ...Equation 2
3. Loop (From Point C to Point D):
In this phase, the block goes through a circular path in the loop.
At point C, the block has both tangential velocity (v_C) and radial acceleration (a_C).
The sum of these two vectors' magnitudes gives us the net acceleration (a_net).
Using the equation for radial acceleration:
a_C = (v_C²) / r where r is the radius of the loop.
At the highest point of the loop, Point D, the net acceleration is directed towards the center and is equal to the centripetal acceleration:
a_net = v_D² / r where v_D is the velocity of the block at the highest point D of the loop.
For the block to not lose contact with the track when it passes through the top of the loop, the net acceleration at Point D must be less than or equal to the gravitational acceleration (g).
Therefore, we have:
v_D² / r ≤ g ...Equation 3
Now, we have all the necessary equations to solve the problem. Knowing the values of the different parameters such as the mass of the block (m), the coefficient of kinetic friction, the length of the horizontal track (displacement), and the radius of the loop (r), we can substitute these values into the equations and solve for the respective unknowns.