example of force

Force: Problem Set with Solutions

This problem set is designed to accompany the PhysicsLibrary entry force. The exercises are stated first for self-study. Complete, worked solutions follow afterward.

Exercises

Problem 1. Net force from perpendicular forces. Figure 1 shows a ring pulled by two perpendicular forces, F₁ = 6 N to the right and F₂ = 8 N upward. Determine the magnitude and direction of the net force.

Problem 2. Horizontal motion with kinetic friction. A 4.0 kg block is pushed across a horizontal floor by a constant horizontal force of 25 N, as shown in Figure 2. The coefficient of kinetic friction is μ_k = 0.30. Find (a) the Normal force, (b) the kinetic friction force, and (c) the horizontal acceleration of the block.

Problem 3. Static friction on an incline. A 5.0 kg block rests on a 25^∘ incline, as in Figure 3. Assume the block is just on the verge of slipping downward. Find the minimum coefficient of static friction required to keep it at rest.

Problem 4. An ideal Atwood Machine. In the system shown in Figure 4, m₁ = 2.0 kg and m₂ = 3.0 kg. The string and Pulley are ideal. Find (a) the magnitude of the acceleration and (b) the string Tension.

Problem 5. Spring force and simple horizontal motion. A 0.40 kg block is attached to a horizontal spring with spring constant k = 80 N∕m. The spring is stretched by x = 0.15 m and released from rest, as in Figure 5. Find (a) the magnitude of the initial spring force and (b) the initial acceleration of the block.

Problem 6. Centripetal force on a flat curve. A 1200 kg CAR travels at 12 m∕s around a flat circular curve of radius 50 m, as sketched in Figure 6. Find (a) the required centripetal force and (b) the minimum coefficient of static friction required if static friction provides the centripetal force.

Worked solutions

Solution 1. Because the two forces are perpendicular, the magnitude of the net force is found from the Pythagorean theorem:

The direction above the positive horizontal axis is

Therefore the net force is

Solution 2. On a horizontal surface with no vertical acceleration,

The kinetic friction force is

The net horizontal force is

Thus the acceleration is

So the answers are

Solution 3. Resolve the weight into components parallel and perpendicular to the incline:

At the threshold of slipping, static friction takes its maximum value,

Since the block is in equilibrium,

Therefore

Hence the minimum coefficient is

Solution 4. Because m₂ > m₁, mass m₂ moves downward and m₁ moves upward. Applying Newton’s second law to each mass gives

Adding the equations,

so

Now solve for the tension:

Therefore

Solution 5. Hooke’s law gives the magnitude of the spring force:

Initially this is the net horizontal force on the block, so

The spring force and acceleration are directed toward equilibrium. Thus

Solution 6. The required centripetal force is

On a flat curve, the centripetal force is supplied by static friction. The normal force is

Thus the minimum coefficient of static friction is

Therefore

Study notes

These six problems illustrate several core ideas related to force:

• forces add vectorially;
• free-body diagrams help isolate all forces acting on a body;
• Friction forces must be modeled carefully, distinguishing static and kinetic friction;
• Newton’s second law connects force to acceleration;
• spring forces are modeled by Hooke’s law;
• centripetal force is not a new interaction but the inward net force needed for circular motion.

References