We should be able to decompose any vector into perpendicular components now: force, velocity, momentum. Below, please see the simplifications we've made in the past and the corresponding update with components:

Work

We already know: "You do positive work if the force and direction of movement are in the same direction, you do negative work if they are in opposite directions, and you do no work if the force is perpendicular to the change in position."

Update: The work is the displacement times the parallel component of the force; or the force times the parallel component of the displacement.

Momentum

We already know: Forces change momentum. If there is no outside force (like in a collision), the momentum of the system is constant. Or $LaTeX: \sum\vec{p}$ must be the same before and after. If there is a force, the impulse is $LaTeX: d\vec{p} = \vec{F}dt$

Update: Momentum is a vector and needs to be added appropriately. Before and after a collision, you add the momentum vectors and the resultant momentum will be the same: in both magnitude and direction. You can do this graphically, or by components: $LaTeX: \sum p_x$ and $LaTeX: \sum p_y$ must be the same before and after the collision.

Torque

We already know: " $LaTeX: \tau=rF_{\perp},$ where $LaTeX: F_{\perp}$ is the force perpendicular to the radius."

Update: " $LaTeX: \tau=rF_{\perp}=r_{\perp}F,$ where $LaTeX: F_{\perp}$ is the component of the force perpendicular to the radius and $LaTeX: r_{\perp}$ is the component of the radius perpendicular to the force.

Statics

We already know: " $LaTeX: \sum\vec{\tau}=I\vec{\alpha}=0,$ and $LaTeX: \sum\vec{F}=m\vec{a}=0.$ "

Update: Torque and force are vectors and must be treated accordingly. In order to make the sums zero, you could add them graphically, but people rarely do it. By components: $LaTeX: \sum F_x=m a_x=0,$ $LaTeX: \sum F_y=m a_y=0,$ and $LaTeX: \sum\tau_z=I\alpha_z=0.$

Exercise 7.1.1

A child in a sled pulled by a force of 40 N up and to the right, Exercise 7.1.1. Starting from rest, you pull a 10 kg sled 10 m on low friction ice with a tension of 40 N oriented as shown in the picture.

How much work did you do on the sled? What is the speed of the sled at the end of 10 m?
If the ice actually had a coefficient of friction of 0.15, estimate the amount of thermal energy produced as you pull the sled 10 m. What is the speed at the end of the 10 m?

Exercise 7.1.2

A wheel being pulled by several different forces at different angles, Exercise 7.1.2. The wheel you see at right is 1 m in diameter. I apply a force of 20 N in four different ways as shown.

Please rank the torques provided by each force from smallest torque to greatest torque.
Please estimate the torque provided by each of the four forces.

Exercise 7.1.3

A ball at the end of low mass pole, free to pivot but not slide, suspended by a cable, Exercise 7.1.3. A 10 kg ball at the end of a 3 m low mass pole (free to pivot but not slide at its base) is supported by a cable attached 1 m from the pivot as shown at right.

Please find the tension in the cable.
Please find the force the wall exerts on the pole, including direction.
How would you modify the solution if the pole also has a mass of 10 kg?

Exercise 7.1.4

A truck and a car heading toward a collision, Exercise 7.1.4. On a very slippery road, a 1000 kg car and a 2000 kg truck have a collision and stick to each other as shown at right. Use drawings:

Could the wreckage have gone off in this $LaTeX: \Downarrow$ direction? How do you know?
Identify the general direction that the wreckage went after collision.
Estimate the final velocity of the wreckage immediately after the collision.
Find the impulse that the car received during the collision. Include direction.
If the collision lasted 1/15 of a second, estimate the average force on the car during the collision. Include direction.