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**Net force** is the unbalanced force acting on an object. When two or more forces act on an object, the resultant (vector sum) of the forces is the net force. In this free body diagram, the net force is to the right, so the object will accelerate to the right.

Most dynamics problems are based on one or a combination of **Newton's Laws of Motion**:

1. **Newton's First Law of Motion (Law of Inertia**): Every body continues in its state of rest or of uniform speed in a straight line unless it is compelled to change that state by a net force acting on it.

2. **Newton's Second Law of Motion**: The acceleration of an object is directly proportional to the **net force** acting on it and is inversely proportional to its mass. The direction of the acceleration is in the direction of the applied net force.

3. **Newton's Third Law of Motion**: Whenever one object exerts a force on a second object, the second exerts an equal and opposite force on the first.

1.1 A 4kg object is moving across a frictionless surface with a constant velocity of 2 m/s. Determine the force necessary to maintain this state of motion

*ANSWER*

zero

Applying Newton's First Law of Motion, this object will continue its state of uniform speed in a straight line until a net (unbalanced) force acts upon it.

Applying Newton's Second Law of Motion, since acceleration is zero, the net force acting on the object is zero. An additional force applied to the object will cause an acceleration.

1.2 An object sits on a frictionless surface. There is a 16 N force being applied to an object and its acceleration is at 2 m/s/s. What is its mass?

*ANSWER*

Applying Newton's Second Law of Motion,

F = ma

16 N = m( 2 m/s^{2})

m = 8 kg

(related search terms: f=ma)

A sledge hammer delivers 4000 N of force to a rock as the rock crumbles. What force does the rock exert on the sledge hammer?

a. less than 4000 N

b. 4000 N

c. more than 4000 N

d. need more information

*ANSWER TO QUICK QUESTION*

b. is correct. See Newton's Third Law above.

Force diagrams are not to be confused with free-body diagrams.

1. The diagram looks down on a marble rolling to the right along a horizontal level table top. The marble is struck by a quick force, F, perpendicular to its velocity. Draw an arrow representing its final velocity.

*ANSWER TO EXAMPLE 1*

Any angle between the initial velocity and the force depending on the magnitudes of the initial velocity and the applied force.

(See below for answers)

1. The diagram is a view looking down on a puck tied to a string and traveling clockwise in a circle on a slippery surface. Draw the path of the puck if the string breaks at this point in the motion.

2. A rocket drifting sideways in space starts its engine and receives a sustained force toward the front of the rocket. Complete the following by drawing the path of the rocket while the force is applied.

3. A **pendulum** is pictured swinging through positions 1 to 5. Complete the picture by drawing the force vectors and net force on the bob at each position.

4. In the following diagram we are looking down on a track with two semicircular sections and two straight sections. A sprinter is running in a counterclockwise direction with constant speed. At each position of the sprinter, draw the following vectors: velocity, acceleration.

5. Draw a force diagram of a 5-strand pulley system.

6. A hunter whirls a stone attached to a string in a horizontal circle above his head. Where should he release the string to hit the target?

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### B. Collinear Force Problems

#### EXAMPLES

#### PROBLEMS

Here "collinear" means the force is in line with motion. The force is either in the same direction as the motion, or in the opposite direction.

2. A 15.0 kg mass pulled along a frictionless surface by a horizontal force of 100 N will have what acceleration?

*ANSWER TO EXAMPLE 2*

**F**_{net} = m**a**

(100 N) = (15.0 kg)**a**

2.1 Two forces act on a body of mass of 2 kg. One is 10N North and the other 4N South. Find resultant acceleration.

*ANSWER TO EXAMPLE 2.1*

**F**_{net} = **F**_{1} + **F**_{2}+ . . .

**F**_{net} =(10 N [North])+(4 N [South]) = 6 N [North]

**F**_{net} = m**a**

6 N [North]= (2 kg) **a**

**a** = 3m/s/s [North]

(See below for answers)

1. A 1550-kg car moving at 10 m/s accelerates to 30 m/s in 10 s. What is the average net force applied to the car?

2. Why, when you release an inflated, untied balloon, does it fly across the room?

3. A horizontal force of 10N is applied to a 30 kg cart on a level floor.

a. How far will it move in 3.0s, starting from rest? Ignore friction.

b. How far will it move in 3.0s if 30N of groceries is added to the cart first?

4. A 6000 kg tractor rests on a flatbed, held in place by chains. The chains provide a maximum horizontal force of 8000 N. When the flatbed is traveling 20 m/s, what is the minimum stopping distance if the chains are not to break?

5. If a force accelerates 4.5 kg at 40 m/s^{2}, that same force would accelerate 18 kg by how much?

6. A mass, m_{1}, accelerates at 3 m/s/s when a force, F, is applied. A second mass, m_{2}, accelerates at 1.0 m/s/s when F is applied to it.

a. Find the value of the ratio m_{1}/m_{2}.

b. Find the acceleration of the combined mass ( m_{1} + m_{2}) under the action of the force, F.

7. An object of mass 2.0 kg starts from rest and slides down an inclined plane 80 cm long in 0.50 seconds. What net force is acting on the object along the incline?

8. A 0.741-kg squash starting from rest achieves a speed of 11 m/s during a time interval of 5.2 ms. What force acted on the squash?

9. A 5.0 kg missile is projected from rest to a speed of 4.5 x 10^{3} m/s. How much time is required to do this by a net force of 5.3 x 10^{5} N?

10. A hockey stick travels backward 11.0 cm while intercepting a .140 kg puck. How much force did the stick exert on the puck to bring the puck from 35.0 m/s to rest?

11. A chain is suspended vertically with the lower end barely touching a tabletop. The 1 kg chain is 1 m long. Upon releasing the chain, each successive link lands and comes to rest on the tabletop. What force does the tabletop exert on the chain when 0.1 m of chain has landed?

12. A 1000-kg car moving north at 100km/h brakes to a stop in 50m. What are the magnitude and direction of the force?

13. Two identical bows shoot arrows straight down. One arrow is 360 grains (about 23.3 g) and the other is 480 grains (about 31.1 g). Which arrow reaches the ground first?

BACK TO TOP OF PAGE(Net force problems with angles.)

3. A 25.0 kg block is pulled along a frictionless horizontal surface by a string. The string makes an angle of 30º to the horizontal and is pulled by a 100 N force. What is the acceleration of the block?

*ANSWER TO EXAMPLE 3*

Sum the forces (The vertical forces cancel as there is no acceleration along the vertical axis.)

**F**_{net} = T_{h} = Tcosø

**F**_{net} = (100 N)cos(30º)

**F**_{net} = 86.6 N

Newton's second law:

**F**_{net} = m**a**

86.6 N = (25.0 kg)**a**

**a** = 3.64 m/s^{2}

1. What is the net force on a puck if a force of 200 N [N] and a force of 300 N [E] is applied to the puck simultaneously?

2. A 4.58 kg crate is at rest on a level icy surface. A cord suddenly exerts a force F = 13.0 N at an angle of 15.5° above the horizontal. At 3.3 seconds what is the crate's speed?

3. An object of mass m = 5.95 kg has an acceleration a = (1.17 m/s/s)x + (-0.664 m/s/s)y. Three forces act on this object: F1, F2, and F3. Given that F1 = (3.22 N) x and F2 = (-1.55 N) x + (2.05 N) y, what is F3?

4. Why is the height of the outer edge of a round track higher than the inner edge?

5. Two horses pull horizontally on ropes attached to a tree. Each horse pulls with a force of magnitude F. If the angle between 2 ropes is 126 degrees, what is the resultant force?

BACK TO TOP OF PAGE4. Consider what happens when a small car collides with a heavy truck. Does the truck exert more force on the car, or does the car exert more force on the truck?

*ANSWER TO EXAMPLE 4*

Neither. They both exert the same amount of force on each other (Newton's Third Law). The car's acceleration is more dramatic because the same force is being applied to a smaller mass.

See below for answers.

1. A father pushes his little girl on a sled.

a. As they accelerate from rest, which force is larger: the force of the little girl on her father or the force of the father on the little girl?

b. Once they are traveling at a constant velocity, which force is larger: the force of the father on his daughter, or the force of friction on the little girl?

c. As they slow down what is the larger force: which force is larger: the force of the father on his daughter, or the force of friction on the little girl?

2. Three blocks are in contact with each other on a frictionless horizontal surface. The masses of the blocks are m_{1 }= 1 kg , m_{2} = 2 kg, m_{3} = 3 kg. A horizontal force F = 24 N is applied on m_{1}.

a. Find the net force on each block.

b. Find magnitude of the contact forces between the blocks.

3. Two blocks, joined by a string, have masses of 6.0 and 9.0 kg. They rest on a frictionless horizontal surface. A 2nd string, attached only to the 9-kg block, has horizontal force = 30 N applied to it. Both blocks accelerate. Find the tension in the string between the blocks.

4. Consider an electron (charge = -1) in orbit around an alpha particle (charge = +2). What is the relationship between the force exerted by the alpha particle on the electron and the force exerted by the electron on the alpha particle?

BACK TO TOP OF PAGESelected solutions are printed below.

For solutions to all the problems on this page click here

C5.

The free body diagram shows the relationship between the two forces.

The vector addition diagram shows the resultant.

Applying the cosine law,

F_{net} = √(F^{2} + F^{2} - 2FFcos54º)

F_{net} = (0.908)F

Since the triangle is isosceles, 180 = 54 + 2θ

So θ = 63º

The resultant force is 0.908F 63º away from either force.

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