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.
To calculate the work done by the ant in pulling the fruit up the hill, we can use the formula:
Work = Force x Distance x cos(theta)
where:
- Force is the net force applied by the ant (the force needed to overcome friction)
- Distance is the distance over which the ant applies the force
- Theta is the angle between the direction of the force and the direction of the displacement (in this case, the angle of the sloped hill)
First, let's find the net force acting on the fruit. We know that the ant applies a force to overcome the kinetic friction:
Friction force = coefficient of kinetic friction x normal force
The normal force is the force exerted by the hill perpendicular to its surface. It can be calculated using the equation:
Normal force = mass x gravitational acceleration x cos(theta)
Since we are given the weight of the fruit, we can divide it by the gravitational acceleration to get the mass (mass = weight / gravitational acceleration). In this case, since only the vertical component of the weight matters, we can use:
Weight = m x g x sin(theta)
Now we can find the normal force and the friction force:
Normal force = m x g x cos(theta)
Friction force = coefficient of kinetic friction x Normal force
Next, let's find the net force needed to pull the fruit up the hill:
Net force = Friction force - Weight
The net force is the force applied by the ant to overcome friction and support the weight of the fruit. Note that the weight acts in the opposite direction as the force applied by the ant.
Now, we can calculate the work done by multiplying the net force by the distance and the cosine of the angle:
Work = Net force x distance x cos(theta)
Let's plug in the values:
Given:
Force (net force) = Friction force - Weight
Distance = 8.60 cm = 0.086 m
Theta (angle) = 16.2°
Coefficient of kinetic friction = 0.540
Weight = 0.00955 N (given)
1. Calculate the mass:
Weight = m x g x sin(theta)
0.00955 N= m x 9.8 m/s^2 x sin(16.2°)
=> m = 0.00955 N / (9.8 m/s^2 x sin(16.2°))
2. Calculate the normal force:
Normal force = m x g x cos(theta)
3. Calculate the friction force:
Friction force = coefficient of kinetic friction x Normal force
4. Calculate the net force:
Net force = Friction force - Weight
5. Calculate the work done:
Work = Net force x distance x cos(theta)
By following these steps, you can calculate the work done by the ant in pulling the piece of fruit up the hill.
W = F * d = 9.55*10^-3 * 8.6*10^-2=
82.13*10^-5 = 8.21*10^-4 Joules.