Final answer:
Work is the dot product of the force vector and displacement vector, and the work done can be calculated by multiplying the respective components of the vectors. Example calculations provided show how to calculate the work for various force and displacement scenarios.
Step-by-step explanation:
Work done by a force on an object is calculated using the formula W = ⟷F ∙ d, where ⟷F is the force vector and d is the displacement vector of the object. This involves taking the dot product of the force and displacement vectors.
1. For the force given by F₁ = (3 N)Î + (4 N)Í, and the displacement from the coordinate (0 m, 0 m) to (5 m, 6 m), we calculate the work done:
- Work, W = ⟷F ∙ ⟷d = (3 N)(5 m) + (4 N)(6 m) = 15 J + 24 J = 39 J.
2. With the force depending on the particle's position given by F₁ = (2y)i + (3x)j, we can only calculate the work done if the path is specified. As for a straight line path along the x-axis from the origin to (5 m, 0), the work done is zero because the force always has a y component only.
Now, let's use these concepts to solve the examples:
- 31. A 20-N constant force pushes a ball over a distance of 5.0 m in the direction of the force. Work done is W = F ⋅ d = 20 N ⋅ 5.0 m = 100 J.
- 32. A cart is pulled by a 20 N force at an angle of 37° to the horizontal over a distance of 6.0 m. To find work, resolve the force into horizontal component (20 N ⋅ cos(37°)) and multiply by the distance. The result is the work done.
- 33. To calculate the work done by the frictional force, you need to know the frictional force, which equals the coefficient of kinetic friction times the normal force. Assuming normal force equals weight (mg), then work done by friction is − Fₕₖ×d (it's negative because friction acts opposite to the displacement).