Q37 PE - College Physics (Urone)

Question

Consider two people pushing a toboggan with four children on it up a snow-covered slope. Construct a problem in which you calculate the acceleration of the toboggan and its load. Include a free-body diagram of the appropriate system of interest as the basis for your analysis. Show vector forces and their components and explain the choice of coordinates. Among the things to be considered are the forces exerted by those pushing, the angle of the slope, and the masses of the toboggan and children.

Step-by-Step Solution

Answer

The forces in the free-body diagram are the forces applied by the people and the weight of the toboggan and its load. The acceleration is calculated as $0.49m/s^2$.

Construct the problem

The acceleration of the toboggan and its load have to be calculated. In order to construct a problem, need to mention the values of the mass of the toboggan and children, the applied force, and the angle of slope. Give the values for these quantities and construct the problem. It can be done as follows:

Consider two people pushing the toboggan with four children having a mass of  $150kg$ by applying a force of $180 N$. If the angle of slope is $45°$, calculate the acceleration of the toboggan and its load.

Draw the free body diagram and calculate acceleration

A free-body diagram can be drawn by considering the forces that act on the toboggan. The force,   applied by the people is acting in the positive x direction. Toboggan is also accelerating in the positive direction. There is an angle of slope,  . The weight of the toboggan and children,  has two components. One is along the negative x direction and the other is along the negative y direction. Thus, the free body diagram can be drawn as follows:

The total force acting along the x direction is equal to the sum of the applied force and the x component of the weight.

 $$\begin{gathered} F_x=m{a}_x \\ F-Wsin\theta =m{a}_x \\ F-mg.sin\theta =m{a}_x \\ {a}_x=\frac{F-mg.sin\theta }{m} \end{gathered}$$

Here,  $F_x$ is the total force acting on the elevator in the x direction and  $g$ is the acceleration due to gravity which is equal to $9.8m/s^2$. The values of applied force, angle of slope and mass are given.

 $$\begin{gathered} F=180N \\ m=150kg \\ \theta =45° \end{gathered}$$

Substitute these values to calculate the acceleration along x axis.

 $$\begin{gathered} a_x=\frac{180-\left(150kg\times sin\left(45°\right)\right)}{150kg} \\ =0.49m/s^2 \end{gathered}$$

Therefore, the acceleration of the toboggan and its load is $0.49m/s^2$.