A hardworking ant must supply 0.00955 N to pull a small piece of fruit at constant velocity over a distance of 8.60 cm up a small hill to the entrance of the colony. If the coefficient of kinetic friction between the piece of fruit and the 16.2° sloped hill is 0.540, calculate the work done by the ant by pulling the piece of fruit up the hill.

Use the work-energy theorem to calculate the mass of the piece of fruit in grams. (The acceleration due to gravity is 9.81 m/s2.)

Well, well, well, looks like our hardworking ant has quite the task on its tiny hands (or should I say, tiny legs!). Let's see if we can help it out with some calculations.

First, let's find the force of friction opposing the ant's efforts. We can use the formula:

Force of friction = coefficient of friction * normal force,

where the normal force is equal to the weight of the object. In this case, the weight can be found by multiplying the mass of the fruit by the acceleration due to gravity (9.81 m/s^2).

Now, the ant is pulling the fruit at a constant velocity, which means the net force acting on it is zero. So we have:

Force of friction + force applied by the ant = 0.

Now, rearranging the equation to solve for the force applied by the ant:

Force applied by the ant = - Force of friction.

Negative sign because the force of friction is acting in the opposite direction to the force applied by the ant.

Given that the force applied by the ant is 0.00955 N and the coefficient of kinetic friction is 0.540, we can substitute those values into our equation:

0.00955 N = - 0.540 * (mass * 9.81 m/s^2).

Now, solving for the mass of the fruit:

mass = (0.00955 N) / (-0.540 * 9.81 m/s^2).

And there you have it! The mass of the piece of fruit in grams can be calculated using the formula above. Just be sure to convert the units accordingly.

Keep up the good work, little ant! You're doing great *applauds*.

To calculate the work done by the ant in pulling the piece of fruit up the hill, we need to determine the force required and the distance traveled.

1. Convert the force required to Newtons (N):
The force required to pull the fruit is given as 0.00955 N.

2. Convert the distance traveled to meters (m):
The distance traveled up the hill is given as 8.60 cm, so we convert it to meters by dividing by 100:
Distance = 8.60 cm / 100 = 0.0860 m.

3. Calculate the work done by the ant:
The work done (W) can be calculated using the formula:
W = force x distance.
W = 0.00955 N x 0.0860 m = 0.0008203 N·m.

Therefore, the work done by the ant in pulling the piece of fruit up the hill is approximately 0.0008203 N·m.

Now, let's calculate the mass of the piece of fruit using the work-energy theorem:

1. Convert the angle of the sloped hill to radians:
The angle of the hill is given as 16.2°. We convert it to radians by multiplying by π/180:
θ = 16.2° x π/180 ≈ 0.2827 radians.

2. Calculate the gravitational potential energy of the fruit:
The gravitational potential energy (PE) of the fruit at the top of the hill equals the work done by the ant. We can express it as:
PE = m x g x h,
where m is the mass, g is the acceleration due to gravity, and h is the height of the hill.

Since the piece of fruit moves at a constant velocity, there is no change in kinetic energy. Therefore, all the work done by the ant goes into increasing the potential energy of the fruit.

3. Substitute the know values into the equation:
0.0008203 N·m = m x 9.81 m/s² x h.

4. Calculate the height of the hill:
h = Distance x sin(θ).
h = 0.0860 m x sin(0.2827) ≈ 0.0425 m.

5. Rearrange the equation and solve for the mass:
0.0008203 N·m = m x 9.81 m/s² x 0.0425 m.
m = 0.0008203 N·m / (9.81 m/s² x 0.0425 m).

6. Convert the mass to grams:
m = m x 1000 grams/kg.

Substituting the values and solving the equation, we find:
m ≈ 0.0195 kg ≈ 19.5 grams.

Therefore, the mass of the piece of fruit is approximately 19.5 grams.

To calculate the work done by the ant to pull the piece of fruit up the hill, we need to determine the force applied by the ant and the distance over which it acts.

First, let's calculate the force applied by the ant. The force applied by the ant is equal to the sum of the force required to overcome gravity and the force due to friction.

The force required to overcome gravity is the weight of the piece of fruit. We can calculate that using the formula:

weight = mass * acceleration due to gravity

Given that the acceleration due to gravity is 9.81 m/s², we need to convert the weight from newtons (N) to kilograms (kg) to use this formula. We can use the following conversion factor:

1 N = 1 kg * m/s²

So, we have:

weight = 0.00955 N

Now, let's calculate the force due to friction. The force due to friction can be calculated using the formula:

frictional force = coefficient of kinetic friction * normal force

The normal force is the perpendicular force exerted by the slope on the piece of fruit. It can be calculated as:

normal force = weight * cos(θ)

θ represents the angle of the slope, which is given as 16.2°.

Now, we can calculate the normal force:

normal force = 0.00955 N * cos(16.2°)

Next, we can calculate the force due to friction:

frictional force = 0.540 * normal force

Now, we can calculate the total force applied by the ant:

total force = weight + frictional force

Next, we need to calculate the distance over which the force is applied. The distance is given as 8.60 cm, but we need to convert it to meters:

distance = 8.60 cm * (1 m / 100 cm)

Once we have the force and distance, we can calculate the work done by the ant using the formula:

work = total force * distance

Finally, to calculate the mass of the piece of fruit using the work-energy theorem, we need to convert the work done into kinetic energy. The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy:

work = ΔKE = (1/2) * mass * velocity^2

Since the piece of fruit is moving at a constant velocity, its kinetic energy is constant, and the work done is equal to the change in kinetic energy. Therefore:

work = (1/2) * mass * velocity^2

Given the work done (in joules) from the previous calculation, we can rearrange the equation to solve for the mass:

mass = (2 * work) / (velocity^2)

Now you can plug in the values and calculate the work done by the ant pulling the piece of fruit up the hill, as well as the mass of the fruit in grams.