Dynamics

## Summary and Practice Applications of Physics Skills

##### © by William Dietsch 2000

Dynamics addresses the cause of motion.  In general the basic laws of dynamics are attributed to the work of Sir Isaac Newton (b 1642, d1727).  His laws of motion are the basis for the mechanical principles which all physicists must master.

Force is classified as a push or a pull.  Forces are vector quantities, requiring a direction as well as magnitude.  The unit of force used in the System International (metric system) is called the Newton (N), which has the units: Kg m/s2.  The unit of force in the increasingly archaic English system is the Pound.  It often seems to a layman that there are lots of forces in nature, however there are really only four forces in the universe.  Gravitational force is the force exerted by all mass carrying bodies.  The gravitational force is only an attractive force.  Electromagnetism is the name given to the force, which occurs between objects with electric charges or magnetic properties.  Electromagnetic forces can be attractive and repulsive.  The strong nuclear force acts within the nucleus of the atom and is responsible for holding the nucleons together.  The weak nuclear force is the force involved in the decay of the nucleus of the atom.  (Technically we might argue that electromagnetism and the weak force have been unified although they are still often discusses as distinct.  And we suspect a new fourth force may be related to the dark energy.

Newton's first law of motion (the law of inertia) states that a body which has no net force acting on it will remain in a state of constant velocity.  This means that a body at rest will remain at rest, and a body moving in a straight line will move in that straight line at a constant speed, forever.

Newton's second law of motion (the law of acceleration) states that the acceleration of a body is directly proportional to the unbalanced force acting on it and is indirectly proportional to the mass of the body.  Laboratory exercise, which you will do, will illustrate these aspects of the second law.  The equation associated with the second law is: F = ma, where F is the net force (see 2 ¶ below), m is the mass, and a is the acceleration of the body.

Newton's third law of motion (the law of reaction) states that for every action there is an equal but opposite reaction.  More about this law will be discussed when we study momentum.  The main implication for this law is that forces always come in pairs, sometimes acting across long distances.

Net force refers to the unbalanced force acting on a body, causing it to accelerate.  A useful way to keep track of forces is by the use of a free body diagram.  (See procedure below.)

Weight is a force, which acts downward on all bodies.  The weight of a body is the force acting on that body as a result of the gravitational pull of the Earth (or other massive body).  Things have weight on other planets, but we will be considering only terrestrial weight here.  Weight is computed using the second law.  Weight (Fw) = mass (m) x acceleration of gravity (g).  Using the values of the metric system, mass is measured in Kg, weight is measured in N, and g has the average value of 9.80 m/s2.

Friction is a force, which opposes motion in the real world.  Sliding (or kinetic) friction occurs as a result of the interaction of surfaces, which are sliding past each other.  Static friction occurs between surfaces, which are in contact with each other, but not moving.  For any given pair of surfaces, static friction is always larger than sliding friction.  Friction depends on the nature of the surfaces in contact, expressed as a coefficient of friction (μ). The coefficient of friction represents the ratio of the force of friction /normal force.  The coefficient is a pure number less than one.  Friction also depends on the force pressing the surfaces together (called the normal force = FN).  If the body is resting on a horizontal surface, the normal force is equal to the weight of the body.

### Free body diagrams

Free body diagrams are used in dynamics to indicate relevant forces, which act on a body.  From a free body diagram, the net force acting on a body can be found more easily.

#### Procedure:

1. Draw a circle to represent the object
2. Inside the circle, indicate the mass of the object (if it is known).  Remember that weight / g = mass.
3. Use arrows to indicate all relevant forces acting on the object.
4. Indicate the value and direction of the acceleration (if it is known).

#### Examples:

1. A stone with a mass of 12 Kg is suspended by a rope, which exerts an upward force of 150 N.  Compute the acceleration of the rock.

2. A box with a mass of 18 Kg is pulled by a rope through which a force of 48 N acts.  A frictional force of 45 N opposes the motion.  Compute the acceleration of the box.

## Force practice problems

In these problems use g = 9.80 m/s2.
1. When a shot-putter exerts a net force on 140 N on a shot, the shot has an acceleration of 19 m/s2.  What is the mass of the shot?

2. Together a motorbike and rider have a mass of 275 kg.  The motorbike is slowed down with an acceleration of -4.50 m/s2.  What is the net force on the motorbike?  Describe the direction of this force and the meaning of the negative sign.

3. A car, mass 1225 kg, traveling at 105 km/h, slows to a stop in 53 m.  What is the size and direction of the force that acted on the car?  What provided the force?

4. Imagine a spider with mass 7.0 x 10-5 kg moving downward on its thread.  The thread exerts an upward force on the spider of 1.2 x 10-4 N.
1. What is the acceleration of the spider?
2. Explain the sign of the acceleration and describe in words how the thread changes the velocity of the spider. (USE A FREE BODY DIAGRAM)

5. What is the weight of each of the following objects?
1. 0.113 kg hockey puck
2. 108 kg football player
3. 870 kg automobile

6. Find the mass of each of these weights.
1. 98 N
2. 80 N
3. 0.98 N

7. A 52 N sled is pulled across a cement sidewalk at constant speed.  A horizontal force of 36 N is exerted.
1. What is the coefficient of sliding friction between the sidewalk and the metal runners of the sled?
2. Suppose the sled now runs on packed snow.  The coefficient of friction is now only 0.12.  If a person weighing 650 N sits on the sled, what force is needed to slide the sled across the snow at constant speed?

8. The coefficient of sliding friction between rubber tires and wet pavement is 0.50.  The brakes are applied a 750 kg car traveling 30 m/s, and the car skids to a stop.
1. What is the size and direction of the force of friction that the road exerts on the car?
2. What would be the size and direction of the acceleration of the car?
3. How far would the car travel before stopping?

9. If the tires of the car in the previous problem did not skid, the coefficient of friction would have been 0.70.  Would the force of friction have been larger, smaller, or the same?  Would the car have come to a stop in a shorter, the same, or a longer distance?  Use calculations to support your answers.

10. A rubber ball weighs 49 N.  a) What is the mass of the ball?  b) What is the acceleration of the ball if an upward force of 69 N is applied?  (USE A FREE BODY DIAGRAM)

11. A small weather rocket weighs 14.7 N.  a) What is its mass?  b) A balloon carries the rocket up.  The rocket is released from the balloon and fired, but its engine exerts an upward force of 10.2 N.  What is the acceleration of the rocket?  (USE A FREE BODY DIAGRAM)

12. The space shuttle has a mass of 2.0 x 106 kg.  At lift-off the engines generate an upward force of 30 x 106 N.
1. What is the weight of the shuttle?  (What assumptions have you made?)
2. What is the acceleration of the shuttle when launched?
3. The average acceleration of the shuttle during its 10 minute launch is 13 m/s2.  What velocity does it attain?
4. As the space shuttle engines burn, the mass of the fuel becomes less and less.  Assuming the force exerted by the engines remains the same, would you expect the acceleration to increase, decrease, or remain the same?  Explain your reasoning.  (USE A FREE BODY DIAGRAM)

13. A certain sports car accelerates from 0 to 60 mph in 9.0 s (average acceleration = 3.0 m/s2).  The mass of the car is 1354 kg.  The average backward force due to air drag during acceleration is 280 N.  Find the forward force required giving the car this acceleration.  (USE A FREE BODY DIAGRAM)

14. An Atwood's machine has a mass of 1.25 Kg suspended on the left connected to a mass of 1.31 Kg on the right.  Compute the acceleration of the system and the tension in the string.  (USE A FREE BODY DIAGRAM)

15. A 50. Kg sled rests on a surface, which has a coefficient of static friction of 0.30, and a coefficient of sliding friction of 0.10.  How much force is required to start the sled moving?  How much force is needed to keep the sled moving?  How much force is required to accelerate the sled at a rate of 3.5 m/s2?

16. A 2500. Kg car is moving at a speed of 14.0 m/s when the brakes are applied, bringing the car to a halt after skidding for 25.0 m.  Compute the acceleration of the car, the force of friction and the coefficient of friction of the tires.

17. An elevator can carry 20 persons of average weight (75 Kg).  The elevator has a mass of 1000. Kg.  The cable, which lifts the elevator, can exert a maximum force of 29600 N.  Compute the maximum acceleration that the elevator can experience.