A person pulls a toboggan for a distance of 20.0 m along the snow with a rope directed 30.0° above the snow. The tension in the rope is 67.0 N.
To find the horizontal and vertical components of the tension force exerted by the rope, we can use trigonometry.
We start by splitting the tension force into its horizontal and vertical components. The horizontal component represents the force acting parallel to the ground, and the vertical component represents the force acting perpendicular to the ground.
The horizontal component can be found using the equation:
F_horizontal = Tension * cos(θ)
where F_horizontal is the horizontal component of the tension force, Tension is the magnitude of the tension force (67.0 N in this case), and θ is the angle the rope makes with the ground (30.0° in this case).
Plugging in the values, we have:
F_horizontal = 67.0 N * cos(30.0°)
Now we can calculate the horizontal component of the tension force.
F_horizontal = 67.0 N * 0.866
= 57.922 N (rounded to three decimal places)
Therefore, the horizontal component of the tension force is approximately 57.922 N.
To find the vertical component, we use the equation:
F_vertical = Tension * sin(θ)
where F_vertical is the vertical component of the tension force, Tension is the magnitude of the tension force (67.0 N in this case), and θ is the angle the rope makes with the ground (30.0° in this case).
Plugging in the values, we have:
F_vertical = 67.0 N * sin(30.0°)
Now we can calculate the vertical component of the tension force.
F_vertical = 67.0 N * 0.5
= 33.5 N
Therefore, the vertical component of the tension force is 33.5 N.
To summarize, the tension in the rope has a horizontal component of approximately 57.922 N and a vertical component of 33.5 N.