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Dynamics - Net Force and Newton's Laws of Motion

Net force is the unbalanced force acting on an object. When tow or more forces act on an object, the resultant (vector sum) of the forces is the net force. In the free body diagram on the left, 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.

Examples

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 TO EXAMPLE 1.1:

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 TO EXAMPLE 1.2:

Applying Newton's Second Law of Motion,

F = ma

16 N = m( 2 m/s²)

m = 8 kg

(related search terms: f=ma)

A. Force Diagrams

(Not free-body diagrams)

Example

1. The following 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.

Problems

1. The following 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 engine is on.

3. A pendulum is pictured below 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.

B. Collinear Force Problems

(Force in line with motion.)

Example

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 = ma

(100 N) = (15.0 kg)a

a = 6.67 m/s²

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 = F₁ + F₂+ . . .

F_net =(10 N [North])+(4 N [South]) = 6 N [North]

F_net = ma

6 N [North]= (2 kg) a

a = 3m/s/s [North]

Problems

1. You decide to take your own 1550-kg car onto the test track. While moving down the track at 10 m/s, you suddenly accelerate to 30 m/s in 10 s. What is the average net force that you have applied to the car during the 10-s interval?

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

3. A shopper in a supermarket pushes a loaded cart with a horizontal force of 10 N. The cart has a mass of 30kg.

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

b. How far will it move in 3.0s if the shopper places their 30N child in the cart before they begin to push it?

4. A load of steel of mass 6000 kg rests on a flatbed of a truck. It is held in place by metal brackets that can exert a maximum horizontal force of 8000 N. When the truck is traveling 20 m/s, What is the minimum stopping distance if the load is not to slide forward?

5. If a force accelerates 4.5 kg at 40 m/s², that same force would accelerate 18 kg by how much?

6. A force F applied to a mass m₁ produces acceleration of 3 m/s/s. The same force applied to a second

mass produces acceleration of 1.0 m/s/s.

a. Find the value of the ratio m₁/m₂.

b. If m₁ and m₂ are combined find the acceleration of the combined mass ( m₁ + m₂) under the action of the

force, F?

7. In a grocery store, you push a 10.9-kg shopping cart with a force of 15.2N. If the cart starts at rest, how far does it move in 3.78s?

8. A 0.741-kg billiard ball is given a speed of 11 m/s during a time interval of 5.2 ms. What force acted on the ball?

9. Scientists are experimenting with a kind of gun that may eventually be used to fire payloads directly into orbit. In one test, this gun accelerates a 5.0 kg projectile from rest to a speed of 4.5 x 10³ m/s. The net force accelerating the projectile is 5.3 x 10⁵ N. How much time is required for the projectile to come up to speed?

10. A .140 kg baseball traveling 35.0 m/s strikes the catcher's mitt, which, in bringing the ball to rest, recoils backward 11.0 cm. what was the average force applied by the ball on the glove?

11. A flexible chain of length 1 m, and mass 2 kg is held from one end so that it hangs vertically, the lower end just touching the surface of a table. The upper end is suddenly released so that the chain falls onto the table and coils up in a small heap, each link coming to rest as soon as it strikes the table. What is the force (in newtons) exerted by the table on the chain when 0.1 m of chain have landed on the table?

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. A lorry of 2500 kg is loaded with 800 kg of materials that need to be transported a distance of 0.6 km. After loading, the lorry accelerates over a distance of 80 m to attain a velocity of 30 mph. The vehicle subsequently maintains constant velocity before decelerating to rest at a rate of 20 m/s/s. Find: the force to accelerate the vehicle and the retarding force to bring it to rest. Also the total time taken during this period and the distance traveled at a constant velocity.

14. 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?

15. A spaceship approaching an unexplored solar system begins slowing down. Consider the following statements:

a. A force is being applied opposite to its direction of motion.
b. A force is being applied in the direction of its motion.
c. The spaceship is slowing down because there is no force keeping it going.
d. The spaceship is slowing down because it is getting closer to a large star.

Which of the above statements is definitely true? Which of the above statements could be true, but does not have to be true?

C. 2-Dimensional Force Problems

(Net force problems with angles.)

Example

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

Free-body diagram:

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 = ma

86.6 N = (25.0 kg)a
a = 3.64 m/s²

Problems

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 block located on a horizontal frictionless floor is pulled by a cord that exerts a force F = 13.0 N at an angle of 15.5° above the horizontal. What is the speed of the block 3.3 seconds after it starts moving?

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.

A chair attached to a vertical rotating pole by two cables, is spun in a horizontal circle at a constant speed. The speed is sufficient to create tension in both the upper and lower cables. The tension in the upper cable is 3500N, L=10 m, and the mass of the chair is 134kg.

a. Draw a free body diagram of the forces on the chair.

b. Determine the radius of the circle and θ.

c. Predict which tension force is greater, T1(tension in the upper cable) or T2 (tension in the lower cable).

d. What is the tension in the lower cable?

6. 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?

D. Force Problems Involving Two or More Objects

Example

4. 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.

Problems

1. A professional wrestler pushes his small son on a sled from behind along a horizontal surface.
a. As they accelerate from rest, which force is larger: the force of the boy on the wrestler or the force of the wrestler on the boy?
b. Once they are traveling at a constant velocity, which force is larger: the force of the wrestler on his son, or the force of friction on the boy?
c. As they slow down what is the larger force: which force is larger: the force of the wrestler on his son, or the force of friction on the boy?

2. Three blocks are in contact with each other on a frictionless horizontal surface.The masses of the blocks are m₁ = 1 kg , m₂ = 2 kg, m₃ = 3 kg. A horizontal force F = 24 N is applied on m₁.

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.

ANSWERS

For solutions to all the problems on this page click here.

C6.

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

The vector addition diagram shows the resultant.

Applying the cosine law,

F_net = √(F² + F² - 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.

For solutions to all the problems on this page click here.