Starting from rest at a height equal to the radius of the circular track, a block of mass 25 kg slides down a quarter circular track under the influence of gravity with friction present (of coefficient μ). The radius of the track is 15 m.

The acceleration of gravity is 9.8 m/s2 .
Determine the work done by the conservative forces. Answer in units of J.

If the kinetic energy of the block at the bottom of the track is 1700 J, what is the work done against friction? Answer in units of J.

Work by conservative force (force of gravity is

W = m•g•h = 25•9.8•15 = = 3675 J.
KE = 1700 J.
W(fr) = W - KE = 3675 - 1700 = 1975 J.

Oh boy, we've got some physics action going on here. Let's jump right in, shall we?

First, let's determine the work done by the conservative forces. Since the block is sliding down a circular track, the only conservative force acting on it is gravity. At the bottom of the track, the block is at its lowest point, which means its potential energy is zero. Therefore, all the potential energy has been converted into kinetic energy.

The work done by the conservative forces (i.e., gravity) is equal to the change in potential energy of the block. Since the potential energy at the top of the track is given by mgh (mass × acceleration due to gravity × height), and the potential energy at the bottom is zero, the work done by gravity is mgh.

In this case, the mass of the block is 25 kg, the acceleration due to gravity is 9.8 m/s², and the height is equal to the radius of the track, which is 15 m. Plugging in these values, we get:

Work done by gravity = (25 kg) × (9.8 m/s²) × (15 m) = 3675 J.

Now, let's move on to the work done against friction. We know that the kinetic energy of the block at the bottom of the track is 1700 J. The work done against friction is equal to the change in kinetic energy of the block.

Since the block is sliding down the track, there's no work done by friction during the descent. Therefore, all the work against friction happens between the bottom of the track and the top. The work done against friction will consume some of the initial kinetic energy, resulting in a smaller final kinetic energy at the bottom of the track.

To find the work done against friction, we need to find the difference between the initial kinetic energy and the final kinetic energy. So, the work done against friction is:

Work against friction = Initial kinetic energy - Final kinetic energy
= 1700 J - 0 J
= 1700 J.

And there you have it! The work done by the conservative forces (gravity) is 3675 J, and the work done against friction is 1700 J.

To determine the work done by conservative forces, we need to calculate the change in potential energy as the block slides down the quarter circular track.

The potential energy of the block at the top of the track is given by:

Potential Energy = m * g * h

where m is the mass of the block, g is the acceleration due to gravity, and h is the height equal to the radius of the track.

Potential Energy = 25 kg * 9.8 m/s^2 * 15 m
Potential Energy = 3675 J

Since the block starts from rest, its initial kinetic energy is zero. Therefore, the change in potential energy is equal to the final kinetic energy.

Change in Potential Energy = Kinetic Energy

So, the work done by the conservative forces is 1700 J.

To determine the work done against friction, we can use the work-energy theorem, which states that the work done by all forces acting on an object is equal to the change in its kinetic energy.

Work against friction = Change in Kinetic Energy - Work by conservative forces

Work against friction = 1700 J - 1700 J

Therefore, the work done against friction is 0 J.

To determine the work done by the conservative forces, we need to consider the gravitational potential energy and the kinetic energy of the block.

At the top of the track, the block has only gravitational potential energy.

Gravitational Potential Energy:
The gravitational potential energy of an object is given by the formula: U = mgh
Where U is the potential energy, m is the mass, g is the acceleration due to gravity, and h is the height above some reference point.

In this case, the height of the object above the reference point is equal to the radius of the circular track, which is 15 m.

U = mgh = (25 kg)(9.8 m/s^2)(15 m) = 3675 J

Next, we need to determine the kinetic energy at the bottom of the track.

The kinetic energy of an object is given by the formula: K = (1/2)mv^2
Where K is the kinetic energy, m is the mass, and v is the velocity of the object.

Given that the kinetic energy at the bottom of the track is 1700 J, we can rearrange the formula to solve for the velocity.

1700 J = (1/2)(25 kg)v^2
v^2 = (2 * 1700 J) / (25 kg)
v^2 = 136 J/kg
v = √136 J/kg

Now, we can find the work done by the conservative forces.

The work-energy principle states that the work done on an object is equal to its change in kinetic energy. In this case, the work done by the conservative forces is equal to the change in kinetic energy.

Work done by the conservative forces = K (final) - K (initial)
Work done by the conservative forces = 1700 J - 0 J (since the block starts from rest)
Work done by the conservative forces = 1700 J

Therefore, the work done by the conservative forces is 1700 J.

To find the work done against friction, we can subtract the work done by the conservative forces from the change in potential energy.

Work done against friction = U (final) - U (initial) - Work done by conservative forces
Work done against friction = 0 J - 3675 J - 1700 J
Work done against friction = -5375 J

However, the work done against friction cannot be negative, as it is the amount of work done by the frictional force. Therefore, the work done against friction is 0 J.