some university students loaded their things into a toboggan. the toboggan has a mass of 8 kg and they are pulling it with a force that is 40 degrees above horizontal. if the coefficient of friction between the toboggan and a dry patch of sidewalk is 0.7,what force is required to pull the toboggan at a constant speed?
To find the force required to pull the toboggan at a constant speed, we need to consider the different forces acting on the toboggan. The main forces to consider are the pulling force and the force of friction.
1. Pulling Force: The force that the students exert on the toboggan is the pulling force. This force can be broken down into horizontal and vertical components.
- The horizontal component of the pulling force is given by F_h = F * cos(40°), where F is the magnitude of the pulling force.
- The vertical component of the pulling force is given by F_v = F * sin(40°).
2. Force of Friction: The force of friction opposes the motion and is given by the equation F_friction = μ * N, where μ is the coefficient of friction and N is the normal force.
- The normal force (N) can be calculated from the weight of the toboggan, which is the mass multiplied by the acceleration due to gravity (N = m * g).
- The force of gravity (weight) is given by W = m * g.
Now, let's calculate the values:
Given:
- Mass of toboggan (m) = 8 kg
- Coefficient of friction (μ) = 0.7
- Angle above horizontal (θ) = 40°
1. Calculate the Forces:
- Calculate the weight of the toboggan (W):
W = m * g = 8 kg * 9.8 m/s² ≈ 78.4 N
- Calculate the normal force (N):
N = W = 78.4 N
- Calculate the force of friction (F_friction):
F_friction = μ * N = 0.7 * 78.4 N ≈ 54.9 N
2. Calculate the Pulling Force:
- Calculate the horizontal component of the pulling force (F_h):
F_h = F * cos(40°)
- Since the toboggan is moving at a constant speed, the pulling force equals the force of friction. Therefore:
F_h = F * cos(40°) ≈ 54.9 N
- Rearrange the equation to solve for F:
F = F_h / cos(40°) ≈ 54.9 N / cos(40°) ≈ 67.0 N
Therefore, the force required to pull the toboggan at a constant speed is approximately 67.0 Newtons.