A person pulls a toboggan for a distance of 43.1 m along the snow with a rope directed 33.3 ° above the snow. The tension in the rope is 109 N. (a) How much work is done on the toboggan by the tension force? (b) How much work is done if the same tension is directed parallel to the snow?
To find the work done by the tension force in both cases, we can use the formula:
Work = Force x Displacement x cos(θ)
where:
- Force is the magnitude of the tension force,
- Displacement is the distance the toboggan is pulled, and
- θ is the angle between the tension force and the direction of displacement.
Let's calculate the work done in each case:
(a) When the tension force is directed 33.3° above the snow:
- Force = 109 N
- Displacement = 43.1 m
- θ = 33.3°
Substituting these values into the formula, we get:
Work = 109 N x 43.1 m x cos(33.3°)
Calculating this expression, we find that the work done by the tension force in this case is approximately 3,936.68 J (Joules).
(b) When the tension force is directed parallel to the snow:
- Force = 109 N
- Displacement = 43.1 m
- θ = 0° (since the force is parallel to the displacement)
Substituting these values into the formula, we get:
Work = 109 N x 43.1 m x cos(0°)
Since cos(0°) = 1, the expression simplifies to:
Work = 109 N x 43.1 m x 1
Calculating this expression, we find that the work done by the tension force in this case is approximately 4,699.90 J (Joules).
Therefore, the work done on the toboggan by the tension force when it is directed 33.3° above the snow is 3,936.68 J, while the work done if the same tension is directed parallel to the snow is 4,699.90 J.
To calculate the work done on the toboggan by the tension force, we can use the formula:
Work = Force × Distance × cos(θ)
Where:
- Work is the amount of work done (in joules, J)
- Force is the applied force (in newtons, N)
- Distance is the distance over which the force is applied (in meters, m)
- θ is the angle between the force and the direction of motion (in degrees)
(a) How much work is done on the toboggan by the tension force?
The given information:
- Force (tension in the rope) = 109 N
- Distance = 43.1 m
- Angle (θ) = 33.3°
Substituting these values into the formula:
Work = 109 N × 43.1 m × cos(33.3°)
Now, we need to calculate the cosine of 33.3°. Keep in mind that most calculators work in radians, so we need to convert the angle to radians first.
33.3° × π / 180° ≈ 0.58 radians
Now, we can calculate the work:
Work ≈ 109 N × 43.1 m × cos(0.58 radians)
Using a calculator:
Work ≈ 4787.08 J
Therefore, approximately 4787.08 joules of work are done on the toboggan by the tension force.
(b) How much work is done if the same tension is directed parallel to the snow?
When the force is parallel to the direction of motion, the angle (θ) between them is 0°. In this case, the cosine of 0° is 1. Therefore, the work done can be calculated as:
Work = 109 N × 43.1 m × cos(0°)
Since cos(0°) = 1:
Work = 109 N × 43.1 m × 1
Work = 4697.9 J
Therefore, in this case, approximately 4697.9 joules of work are done on the toboggan by the tension force.