A father pulls his young daughter on a sled with a constant velocity on a level surface through a distance of 10 m, as illustrated below. If the total mass of the sled and the girl is 35 kg and the coefficient of kinetic friction between the sled runners and the snow is 0.20, how much work does the father do?
610 J
To calculate the work done by the father, we need to determine the net force acting on the sled and the distance it is being displaced.
The net force can be calculated using the equation:
Net Force = Frictional Force + Applied Force
The frictional force can be found using the equation:
Frictional Force = Coefficient of Friction * Normal Force
The normal force is the force exerted by the surface on the sled and is equal to the weight of the sled and the girl which can be calculated by:
Normal Force = Mass * Gravity
Given:
Mass of sled and girl (m) = 35 kg
Coefficient of kinetic friction (μ) = 0.20
Distance (d) = 10 m
Acceleration due to gravity (g) = 9.8 m/s^2
First, calculate the normal force:
Normal Force = Mass * Gravity
Normal Force = 35 kg * 9.8 m/s^2
Normal Force = 343 N
Next, calculate the frictional force:
Frictional Force = Coefficient of Friction * Normal Force
Frictional Force = 0.20 * 343 N
Frictional Force = 68.6 N
Since the sled is moving at a constant velocity, the net force is equal to zero. Therefore, the applied force must be equal in magnitude and opposite in direction to the frictional force:
Applied Force = Frictional Force
Applied Force = 68.6 N
Finally, calculate the work done by the father:
Work = Force * Distance * cos(theta)
Since the sled is being pulled in the same direction as the applied force, the angle between the force and displacement is 0 degrees, and cos(0) is equal to 1.
Work = Applied Force * Distance
Work = 68.6 N * 10 m
Work = 686 Joules
Therefore, the father does 686 Joules of work.
To find out how much work the father does, we first need to understand the definition of work and how it is calculated.
Work is defined as the product of the force applied on an object and the displacement of the object in the direction of the force. Mathematically, it can be expressed as:
Work (W) = Force (F) x Displacement (d) x Cos(θ)
Where:
- W is the work done
- F is the force applied
- d is the displacement of the object
- θ is the angle between the force and the displacement vectors
In this case, the father is pulling the sled, so the force he is applying is in the direction of the displacement. Therefore, the angle θ between the force and the displacement vectors is 0 degrees, and the cosine of 0 degrees is 1, so we can disregard the cosine term.
The work done by the father can be calculated as:
W = F x d
Now, we need to find the force applied by the father. To do this, we need to consider the forces acting on the sled.
1. Gravitational force (Weight):
The weight of the sled and the girl is given by the mass (m) and acceleration due to gravity (g).
Weight (Wg) = m x g
2. Normal force (N):
The normal force exerted on the sled by the surface is equal in magnitude and opposite in direction to the weight of the sled and the girl.
3. Frictional force (f):
The frictional force opposing the motion is given by the coefficient of kinetic friction (μk) multiplied by the normal force.
f = μk x N
Since the sled is moving at a constant velocity, the net force acting on it must be zero. Therefore, the force exerted by the father in pulling the sled must balance the frictional force.
F = f
Now, we can substitute the values and calculate the work.
m = 35 kg (the total mass of the sled and the girl)
g = 9.8 m/s^2 (acceleration due to gravity)
μk = 0.20 (coefficient of kinetic friction)
d = 10 m (displacement)
1. Calculate the weight:
Wg = m x g
2. Calculate the normal force (equal to the weight):
N = Wg
3. Calculate the frictional force:
f = μk x N
4. Calculate the force exerted by the father:
F = f
5. Calculate the work done by the father:
W = F x d
Let's plug in these values and calculate the work done by the father.