A body slides down a curved track which is one quadrant of a circle of radius R. If it starts from rest at the top of the track and there is no friction And the body starts from rest it's speed at the bottom of the track is
To find the speed of the body at the bottom of the track, we can use the conservation of energy principle. At the top of the track, the body has gravitational potential energy, and at the bottom of the track, it has kinetic energy.
The total mechanical energy at any point on the track is the sum of the potential energy and the kinetic energy of the body. Assuming no energy is lost due to friction or other forms of energy dissipation, we can equate the potential energy at the top of the track to the kinetic energy at the bottom of the track.
The potential energy at the top of the track is equal to the gravitational potential energy, which is given by the formula:
PE = mgh
where m is the mass of the body, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height from the top of the track to the bottom.
The kinetic energy at the bottom of the track is given by the formula:
KE = (1/2)mv^2
where v is the speed of the body at the bottom.
Since energy is conserved, we can equate the potential energy at the top (PE) to the kinetic energy at the bottom (KE):
mgh = (1/2)mv^2
The mass of the body cancels out from both sides of the equation:
gh = (1/2)v^2
Solving for v gives:
v = √(2gh)
Therefore, the speed of the body at the bottom of the track is given by the square root of twice the product of the acceleration due to gravity (g) and the height (h) from the top of the track to the bottom.
Note: Since the body starts from rest, its initial velocity is zero, and this calculation assumes that there is no friction or other forces acting on the body as it slides down the track.