# Newton's Laws and Dynamics

Mathematics I · Physics · https://tryals.app/learn/mathematics-i/newtons-laws-and-dynamics

## Forces and Newton's Laws

**Dynamics** explains motion through **forces**, which are pushes or pulls represented as **vectors** with magnitude and direction.

| Law | Definition | Key Rule |
| --- | --- | --- |
| **First Law (Inertia)** | Objects keep constant velocity unless acted on by a net force. | Motion needs no force to continue; only **change** does. |
| **Second Law** | $\vec{F} = m\vec{a}$ (Net force equals mass times acceleration). | It is a differential equation $\vec{F} = m\ddot{x}$ driving trajectories. |
| **Third Law** | For every action, an equal and opposite reaction. | Action-reaction pairs act on **two different objects**. |

The **net force** is the vector sum of all forces governing motion.

## Equilibrium and Pitfalls

An object in **equilibrium** experiences zero net force and zero acceleration, meaning it is either at rest or moving at a constant velocity. Common forces like weight ($mg$), tension, friction, and normal force combine vectorially.

Because forces are **vectors**, resolve them into components along chosen axes to apply Newton's second law component by component.

> **Common Pitfalls:**
> - Thinking a force is required to **keep** an object moving.
> - Misapplying the **third law** by assuming action-reaction pairs cancel out; they act on different objects. Forces that cancel for equilibrium act on the **same** object.

## Practice questions

5 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. Newton's third law says the forces of an action-reaction pair are equal and opposite. Why doesn't every pair of forces cancel, leaving nothing able to accelerate?

A. Because the two forces act on different objects.
B. Because the reaction force is slightly smaller than the action force, leaving a net effect.
C. Because the reaction force arrives a moment later, so the two never coincide in time.
D. Because action-reaction pairs only occur when the objects are in contact, which is rare.

**Answer:** A. Because the two forces act on different objects.

**Why:** The second law is applied to one object at a time, and the reaction force acts on the *other* object, so it never appears on the first one's free-body diagram and cannot cancel anything there. Which is why free-body diagrams are the central technique rather than a study aid: drawing the boundary makes the third-law partner automatically absent.

Page: https://tryals.app/practice/mathematics-i/newtons-laws-and-dynamics/newtons-third-law-says-the-forces-of-an-action-reaction-pair-are

### 2. Sort each situation by whether the NET force on the object is zero or nonzero.

**Answer:**

- Net force is zero: A car cruising at a steady 25 m/s on a straight road, A book resting on a table, A skydiver falling at terminal velocity
- Net force is nonzero: A ball at the top of its flight, A satellite circling the Earth at constant speed, A train braking to a halt

**Why:** Uniform velocity, including the skydiver's terminal velocity, requires no net force at all, motion continues by inertia. The ball, the satellite and the braking train all have changing velocities, so each has an unbalanced force. Assuming that anything moving must be pushed is the pre-Newtonian instinct the first law overturns.

Page: https://tryals.app/practice/mathematics-i/newtons-laws-and-dynamics/sort-each-situation-by-whether-the-net-force-on-the-object-is-zero-or

### 3. A book sits on a table. The table pushes up on the book with exactly the book's weight. Is that Newton's third law?

A. Yes, the two forces are equal and opposite, which is the third law
B. No, these act on the same object and balance under the first law
C. Yes, and it is also an example of the second law with zero acceleration
D. No, because the third law applies only to moving objects

**Answer:** B. No, these act on the same object and balance under the first law

**Why:** Both forces act on the book, so they are a first-law balance, not a third-law pair, the partner of the table's push is the book pushing down on the table. It is the commonest third-law error, and the test is always the same: name the two objects and check they differ.

Page: https://tryals.app/practice/mathematics-i/newtons-laws-and-dynamics/a-book-sits-on-a-table-the-table-pushes-up-on-the-book-with-exactly

### 4. Select every statement that is consistent with Newton's laws.

A. A force is needed to keep an object moving at constant velocity
B. Doubling the net force on a fixed mass doubles its acceleration
C. For a fixed net force, doubling the mass halves the acceleration
D. Steady motion in a straight line means the forces on the object balance
E. You and the Earth pull on each other, so the two pulls cancel and nothing accelerates

**Answer:** B. Doubling the net force on a fixed mass doubles its acceleration; C. For a fixed net force, doubling the mass halves the acceleration; D. Steady motion in a straight line means the forces on the object balance

**Why:** Options 1, 4 and 5 restate the first and second laws. Option 2 is the pre-Newtonian error, force changes motion, it does not sustain it. Option 3 misreads the third law: its two forces act on _different_ bodies, so they can never cancel each other.

Page: https://tryals.app/practice/mathematics-i/newtons-laws-and-dynamics/select-every-statement-that-is-consistent-with-newtons-laws

### 5. Order the standard procedure for solving a dynamics problem.

**Answer:**

1. Draw a free-body diagram showing every force acting on the object
2. Choose axes and resolve each force into components
3. Add the components to obtain the net force along each axis
4. Apply F = ma along each axis to find the acceleration
5. Integrate the acceleration to obtain the velocity and the position

**Why:** The second law is a vector equation, so it is applied one axis at a time, which is why resolving into components comes before summing and before F = ma. Only once the acceleration is known does calculus take over and produce the trajectory.

Page: https://tryals.app/practice/mathematics-i/newtons-laws-and-dynamics/order-the-standard-procedure-for-solving-a-dynamics-problem