Newton's Laws Practice Problems with Fully Worked Solutions

Overview

This guide gives you a compact bank of mechanics practice questions that cover the most common ways Newton’s laws appear in undergraduate physics notes, homework, and exams. The problems move from direct force balances to multi-force systems involving friction, tension, and inclined planes. Each one is written to reinforce a habit that matters more than memorizing isolated formulas.

Before the problems, keep these ideas in view:

• Newton’s first law: if the net force on an object is zero, its velocity is constant. Constant velocity includes the special case of rest.
• Newton’s second law: the net force on an object equals mass times acceleration, written as ΣF = ma.
• Newton’s third law: interaction forces come in equal-magnitude, opposite-direction pairs acting on different objects.

For force problems, the most common mistake is treating all forces in a diagram as though they act on one object. They do not. A free-body diagram should include only forces acting on the chosen object. That single habit clears up many confusing mechanics questions.

If unit conversions slow you down, keep Physics Units, Constants, and Conversions Cheat Sheet nearby. If you want a broader summary of formulas used across topics, College Physics Formula Sheet by Topic: Mechanics, E&M, Thermodynamics, Waves, and Modern Physics is a useful companion.

Problem 1: Net force from a known mass and acceleration

Question: A 4.0 kg cart accelerates to the right at 2.5 m/s² on a horizontal track. What is the net force on the cart?

Solution:

Use Newton’s second law:

ΣF = ma

Substitute the values:

ΣF = (4.0 kg)(2.5 m/s²) = 10 N

Answer: The net force is 10 N to the right.

Why this matters: this is the basic translation between motion and force. If acceleration is known, the net force follows directly.

Problem 2: Balanced forces and constant velocity

Question: A student pushes a box across the floor at constant velocity. The applied force is 35 N to the right. What is the friction force?

Solution:

Constant velocity means acceleration is zero.

So the net force must be zero:

ΣF = 0

If the push is 35 N to the right, friction must balance it with 35 N to the left.

Answer: The friction force is 35 N to the left.

Key idea: motion does not require a net force. Change in motion does.

Problem 3: Finding acceleration with opposing forces

Question: A 6.0 kg sled is pulled rightward with 30 N while friction acts leftward with 12 N. Find the acceleration.

Solution:

Step 1: Find the net force.

ΣF = 30 N - 12 N = 18 N to the right.

Step 2: Apply Newton’s second law.

a = ΣF / m = 18 N / 6.0 kg = 3.0 m/s²

Answer: The acceleration is 3.0 m/s² to the right.

Check: the acceleration points in the direction of the net force, as it should.

Problem 4: Weight and normal force on a level surface

Question: A 12 kg crate rests on a horizontal floor. What are the magnitudes of its weight and normal force?

Solution:

Weight is

W = mg = (12 kg)(9.8 m/s²) = 117.6 N

Because the crate is at rest and there is no vertical acceleration, the normal force balances the weight:

N = 117.6 N

Answer: Weight = 117.6 N downward; normal force = 117.6 N upward.

Common reminder: the normal force is not always equal to weight, but on a level surface with no vertical acceleration, it is.

Problem 5: Elevator problem

Question: A 70 kg passenger stands on a scale in an elevator accelerating upward at 1.5 m/s². What does the scale read?

Solution:

The scale reads the normal force.

Choose upward as positive. Then

ΣF = N - mg = ma

So

N = m(g + a)

Substitute:

N = 70(9.8 + 1.5) = 70(11.3) = 791 N

Answer: The scale reads 791 N.

Interpretation: upward acceleration makes the apparent weight larger than true weight.

Maintenance cycle

This section helps you use the problem bank as a recurring study tool instead of a one-time read. Newton’s laws are learned best through spaced review. Revisit the same skill types under slightly different conditions: more forces, different axes, a different unknown, or a different object selected for the free-body diagram.

A practical maintenance cycle looks like this:

1. Early learning: solve direct one-object problems with horizontal motion and no friction.
2. Skill building: add friction, vertical forces, and apparent weight problems.
3. Exam review: mix in inclined planes, connected blocks, and conceptual questions about Newton’s third law.
4. Refresh cycle: return weekly or before each quiz to redo a few problems without notes.

Here are additional worked examples for that cycle.

Problem 6: Static friction check

Question: A 10 kg box sits on the floor. A horizontal push of 20 N is applied, but the box does not move. What is the static friction force?

Solution:

If the box does not move, then acceleration is zero and the net horizontal force is zero.

Static friction adjusts to match the applied force up to its maximum value.

So the static friction force must be 20 N opposite the push.

Answer: 20 N opposite the applied force.

Lesson: static friction is not automatically equal to μsN. That expression gives the maximum possible static friction, not always the actual value.

Problem 7: Kinetic friction on a horizontal surface

Question: A 5.0 kg block slides on a horizontal table. The coefficient of kinetic friction is 0.20. Find the friction force and resulting acceleration if a 18 N horizontal pull acts on the block.

Solution:

Step 1: Find the normal force.

Since there is no vertical acceleration,

N = mg = (5.0)(9.8) = 49 N

Step 2: Find kinetic friction.

f_k = μ_kN = (0.20)(49) = 9.8 N

Step 3: Find net horizontal force.

ΣF = 18 - 9.8 = 8.2 N

Step 4: Find acceleration.

a = 8.2 / 5.0 = 1.64 m/s²

Answer: Friction force = 9.8 N; acceleration = 1.64 m/s² in the direction of the pull.

Problem 8: Inclined plane without friction

Question: A 3.0 kg block slides down a frictionless 30° incline. Find its acceleration.

Solution:

On an incline, choose axes parallel and perpendicular to the surface. The component of weight along the slope is

mg sin θ

So the net force along the slope is

ΣF = mg sin θ

Then

ma = mg sin θ

The mass cancels:

a = g sin θ = 9.8 × sin 30° = 9.8 × 0.5 = 4.9 m/s²

Answer: The acceleration is 4.9 m/s² down the incline.

If inclined planes still feel awkward, review your axis choices before doing algebra. This is where many students lose points, not in the calculation itself. For motion-based comparisons, Kinematics Equations Explained: When to Use Each Formula pairs well with force analysis.

Problem 9: Atwood machine

Question: Two masses are connected by a light string over a frictionless pulley. Let m₁ = 2.0 kg and m₂ = 5.0 kg. Find the acceleration magnitude.

Solution:

The heavier mass, 5.0 kg, moves downward; the lighter mass moves upward.

Write Newton’s second law for each mass:

For m₂: m₂g - T = m₂a

For m₁: T - m₁g = m₁a

Add the equations to eliminate tension:

m₂g - m₁g = (m₁ + m₂)a

So

a = (m₂ - m₁)g / (m₁ + m₂)

Substitute:

a = (5.0 - 2.0)(9.8) / (7.0) = 29.4 / 7.0 = 4.2 m/s²

Answer: The acceleration magnitude is 4.2 m/s².

Study note: connected-object problems become easier when you first decide the positive direction for each object in a way that matches the actual motion.

Signals that require updates

Even evergreen physics practice articles benefit from occasional revision. The laws themselves do not change, but the way students search for help does. This page should be refreshed on a scheduled cycle and whenever search intent shifts toward a different format, such as more conceptual traps, more exam-style mixed questions, or more free-body diagram problems.

Good signals that this topic needs updating include:

• Students repeatedly miss the same setup step. If readers struggle with axis choice or third-law pairs, add a new example aimed directly at that skill.
• The article has too many direct calculation problems and not enough interpretation. Exams often ask what happens to acceleration when one force changes, not just for a number.
• Common textbook patterns shift. For example, if more instructors emphasize friction limits or multi-step systems, expand those sections.
• The article needs more return value. A maintenance-style piece should give readers a reason to come back, such as a rotating set of challenge problems.

Problem 10: Newton’s third law conceptual check

Question: A truck collides with a small car. During the collision, which exerts the larger force: the truck on the car, or the car on the truck?

Solution:

By Newton’s third law, the forces are equal in magnitude and opposite in direction.

Answer: Neither exerts a larger force. The magnitudes are equal.

Important clarification: equal force does not mean equal damage or equal acceleration. Because the car usually has smaller mass, the same force produces a larger acceleration magnitude on the car.

Problem 11: Two blocks in contact

Question: A 2.0 kg block and a 3.0 kg block sit in contact on a frictionless surface. A 15 N horizontal force pushes the 2.0 kg block, and both blocks move together. Find the acceleration of the system and the contact force between the blocks.

Solution:

Step 1: Treat both blocks as one system.

Total mass = 2.0 + 3.0 = 5.0 kg

Acceleration:

a = F / m = 15 / 5.0 = 3.0 m/s²

Step 2: Find the contact force.

Analyze the 3.0 kg block alone. The only horizontal force on it is the contact force from the 2.0 kg block.

F_contact = ma = (3.0)(3.0) = 9.0 N

Answer: System acceleration = 3.0 m/s²; contact force = 9.0 N.

Update insight: this is a good example to include because it teaches when to treat objects as a single system and when to isolate one part.

Common issues

This section targets the errors that make force problems feel harder than they are. Most Newton’s laws mistakes happen before any arithmetic starts.

1. Skipping the free-body diagram. Even a rough sketch helps separate real forces from imagined ones. Weight, normal force, tension, friction, and applied force should each have a clear source.
2. Using the wrong object. If the question asks about one block, do not include forces acting on another block unless they act through contact or tension on the chosen object.
3. Confusing velocity with acceleration. An object can move right while accelerating left.
4. Assuming normal force always equals weight. This is only true in specific situations.
5. Using friction formulas automatically. Static friction must be checked; kinetic friction applies once sliding occurs.
6. Mixing sign conventions. Pick positive directions once and keep them consistent.

Here is one final worked example that combines several of these ideas.

Problem 12: Inclined plane with friction

Question: A 4.0 kg block is pulled up a 25° incline by a force of 40 N parallel to the incline. The coefficient of kinetic friction is 0.15. Find the acceleration.

Solution:

Choose the positive direction up the incline.

Forces along the incline:

• Applied force up the incline: 40 N
• Component of gravity down the incline: mg sin θ
• Kinetic friction down the incline: f_k = μ_kN

Step 1: Find the normal force.

N = mg cos θ = (4.0)(9.8)cos 25°

N ≈ 39.2 × 0.906 ≈ 35.5 N

Step 2: Find friction.

f_k = 0.15 × 35.5 ≈ 5.33 N

Step 3: Find the component of weight along the slope.

mg sin θ = 39.2 × sin 25° ≈ 39.2 × 0.423 ≈ 16.58 N

Step 4: Write Newton’s second law along the incline.

ΣF = 40 - 16.58 - 5.33 = 18.09 N

Step 5: Solve for acceleration.

a = 18.09 / 4.0 ≈ 4.52 m/s²

Answer: The acceleration is about 4.5 m/s² up the incline.

Why students miss this: the normal force is not equal to mg on an incline, so friction cannot be computed from μmg directly.

When to revisit

Use this article as a recurring check-in point rather than a one-time set of notes. Revisit it when you start forces in class, before homework sets on free-body diagrams, before a mechanics midterm, and again when later topics rely on force analysis. A short return session is often enough: pick two easy problems, two mixed problems, and one conceptual question.

A practical review routine looks like this:

• Once a week: solve one straight calculation problem and one conceptual Newton’s third law problem from memory.
• Before quizzes: redo at least one friction problem and one incline problem without looking at the solution first.
• Before major exams: mix force problems with motion questions so you can connect dynamics to kinematics. The article Kinematics Equations Explained: When to Use Each Formula is useful for that transition.
• When stuck on homework: compare your setup, not just your final number, to the examples here.

If you want to keep improving, build your own mini bank of variants:

1. Change the mass but keep the same structure.
2. Reverse the direction of the applied force.
3. Add or remove friction.
4. Change the unknown from acceleration to tension or normal force.
5. Rewrite one numerical problem as a conceptual prediction question.

That kind of deliberate revisiting is what turns physics homework help into lasting understanding. For continued study support, you can also explore A semester-by-semester roadmap for learning physics online without getting lost. And if you want to apply mechanics ideas in a more hands-on setting, A beginner’s guide to physics lab simulations: experiments you can run from home is a useful next step.

Come back to this page whenever Newton’s laws start to feel abstract again. The best time to revisit is not only before an exam, but also right after you notice a pattern of missed setup steps. In force problems, a small reset in method often does more than another hour of memorizing formulas.