Problem 4: A block of mass m slides down a frictionless incline as: The block is released from height h above the bottom of the circular loop. (a) Find the force exerted on the block by the inclined track at point A. (Hint: Consider the Newton’s 2nd law in the radial direction and you may use polar coordinates for this circular part of the path to relate the acceleration and velocity components). (b) Find the force exerted on the block by the inclined track at point B. (c) Find the speed of the block at point B. (d) What is the maximum height that the block can reach after leaving the track? (e) Find the distance between point A and the point that the block land on ground level?
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To solve this problem, we can break it down into several steps:
Step 1: Analyze the forces acting on the block at each point.
Step 2: Apply Newton's second law to determine the acceleration.
Step 3: Use equations of motion to find the speed at point B.
Step 4: Calculate the maximum height the block can reach.
Step 5: Determine the distance between point A and the point where the block lands on the ground.
Let's go through each step in detail:
Step 1: Forces on the block at each point.
At point A, the only force acting on the block is its weight (mg) acting vertically downwards.
At point B, there are two forces acting on the block: its weight (mg) acting vertically downwards and the normal force (N) acting perpendicular to the incline.
Step 2: Newton's second law at point A.
Since the block is moving on a circular path at point A, we need to consider the radial direction. In polar coordinates, the acceleration can be expressed as a = (a_r, a_θ). As the incline is frictionless, the only force on the block is its weight (mg). Therefore, we can use Newton's second law in the radial direction to relate the acceleration and velocity components.
Step 3: Speed at point B.
To find the speed at point B, we can use the conservation of mechanical energy. The block starts with potential energy at point A, which gets converted to kinetic energy at point B. By equating these energies, we can solve for the speed at point B.
Step 4: Maximum height reached after leaving the track.
To find the maximum height the block can reach after leaving the track, we can again use the conservation of mechanical energy. At the highest point, all of the block's initial mechanical energy is converted to potential energy. By equating the initial mechanical energy with the potential energy at the maximum height, we can solve for the maximum height.
Step 5: Distance between point A and the point where the block lands on the ground.
To find the distance between point A and the landing point on the ground, we can use the equations of motion in the vertical direction. We need to find the time it takes for the block to reach the ground from the maximum height and multiply it by the horizontal component of the block's velocity at the maximum height.
By following these steps, we can solve each part of the problem and find the desired quantities.