a block of a mass starts from rest and slides down a surface which correspnds to a quarter of a circle. if the curved surface is smooth, what is the speed at the bottom?
V = sqrt(2 g R)
where R is the radius of the quarter-circle.
Friction is being assumed zero.
To determine the speed of the block at the bottom of the quarter-circle, we can use the principles of conservation of energy and circular motion.
First, let's understand the situation. A block of mass m starts from rest, which means its initial velocity is zero. It slides down a smooth, curved surface corresponding to a quarter of a circle. The curved surface is smooth, implying that there is no friction or energy loss due to surface roughness.
Since there is no friction, the only force acting on the block is its weight (mg), which points in the vertically downward direction.
Now, let's tackle the problem step by step:
1. Consider the forces acting on the block at different positions. At any point along the curved surface, the normal force (N) acts perpendicular to the surface supporting the block, and its weight (mg) acts vertically downward.
2. At the bottommost point of the quarter-circle, the normal force is perpendicular to the surface and does not contribute to the motion. The weight (mg) acts in the direction of motion.
3. From the conservation of energy, we can say that the total mechanical energy (E) of the system is conserved. Initially, the block has gravitational potential energy (mgh), but as it reaches the bottom, all the potential energy is converted into kinetic energy (1/2 mv^2), where v represents the speed of the block at the bottom.
4. Equating the initial potential energy to the final kinetic energy, we have mgh = 1/2 mv^2. Here, h represents the height of the starting position above the bottom.
5. Canceling the mass (m) from both sides of the equation, we get gh = 1/2 v^2.
6. Now, rearranging the equation, we find v^2 = 2gh.
7. Finally, taking the square root of both sides, we obtain v = √(2gh).
Therefore, the speed of the block at the bottom of the quarter-circle is given by the equation v = √(2gh), where g represents the acceleration due to gravity, and h represents the height of the starting position above the bottom.