The Problem

A body of mass 2kg slides down a curved track, which is a quadrant of a circle with a radius of 1m. All surfaces are frictionless. The body starts from rest, and we need to find its speed at the bottom of the track.

Understanding the Situation

To solve this problem, we need to apply the principles of conservation of energy and circular motion. Let's break down the solution into the following steps:

1. Determine the gravitational potential energy at the top of the track.
2. Calculate the gravitational potential energy at the bottom of the track.
3. Apply the principle of conservation of energy to find the speed at the bottom.

Determining the Gravitational Potential Energy at the Top

At the top of the track, the body is at a height of 1m above the ground. The gravitational potential energy (PE) at this height is given by the formula:

PE = m * g * h

where m is the mass of the body (2kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height (1m).

Plugging in the values, we get:

PE_top = 2kg * 9.8 m/s² * 1m
= 19.6 J

Calculating the Gravitational Potential Energy at the Bottom

At the bottom of the track, the body is at ground level, so its height is 0m. Therefore, the gravitational potential energy at the bottom is:

PE_bottom = m * g * h
= 2kg * 9.8 m/s² * 0m
= 0 J

Applying the Conservation of Energy

According to the principle of conservation of energy, the total mechanical energy of the system remains constant throughout the motion. In this case, the total mechanical energy is the sum of the kinetic energy (KE) and the gravitational potential energy (PE).

At the top of the track, the body starts from rest, so its initial kinetic energy is 0 J. Therefore, the total mechanical energy at the top is equal to the gravitational potential energy:

Total Energy_top = KE_top + PE_top
= 0 J + 19.6 J
= 19.6 J

At the bottom of the track, the gravitational potential energy is 0 J. Therefore, the total mechanical energy at the bottom is equal to the kinetic energy:

Total Energy_bottom = KE_bottom + PE_bottom
= KE_bottom + 0 J
= KE_bottom

Since the total mechanical energy is conserved, we can equate the total energy at the top to the total energy at the bottom:

Total Energy_top = Total Energy_bottom

19.6 J = KE_bottom

Finding the Speed at the Bottom

The kinetic energy (KE) is given by the formula:

KE = (1/2) * m * v²

where m is the mass of the body (2kg) and v is the velocity.

We can rearrange the equation to solve for v:

v = √(2 * KE / m)

Plugging in the values, we get:

v = √(2 * 19.6 J / 2kg)
= √(19.6 J / 2