Posible Prblem 4

Newton’s Second Law and the Friction and Spring Forces Quick review of Newton’s 2nd law Static friction

Drawing and using free body diagrams

Kinetic friction

Resolving vectors into x and y components

Springs

Sample problems

Newton’s 2nd law in a larger context

Monday, June 8, Session 1 N. J. Smith, MTSU, June 2009

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Newton’s Second Law

The sum of the forces acting on an object is equal to the product of the object’s mass and its acceleration. N. J. Smith, MTSU, June 2009

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Newton’s second law is a vector equation. Vectors have both magnitude and direction. The magnitude of a vector is represented symbolically by the same symbol as the vector, but without the arrow:

Vector magnitudes are always positive.

N. J. Smith, MTSU, June 2009

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The directions of the acceleration of an object and the net force applied are the same.

N. J. Smith, MTSU, June 2009

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Static Friction y

You push against a filing cabinet, but it doesn’t move.

x

N. J. Smith, MTSU, June 2009

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The static friction force can have a magnitude from zero, up to some maximum value. i.e. fs ranges from 0 to fs, max The static friction force acts in a direction opposite to the direction of motion the object would have if there was no friction. The magnitude of the maximum static friction force is experimentally found to be related to the normal force:

This is not a vector equation.

N. J. Smith, MTSU, June 2009

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Example 1

Solution

If the coefficient of static friction between the filing cabinet and the floor is 0.92, what is the maximum force that can be applied before the cabinet begins to slide? The mass of the cabinet is 120 kg.

N. J. Smith, MTSU, June 2009

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Kinetic Friction What happens if a force greater than the filing cabinet?

is applied to

We find, again through experiment, that the frictional force does not disappear, but assumes a constant value, which is smaller than the static friction. The frictional force present between two objects sliding past each other is called the kinetic friction force, fk. The kinetic frictional force is related to the normal force by the coefficient of kinetic friction, µ k : This is not a vector equation.

N. J. Smith, MTSU, June 2009

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Example 2

Solution

The applied force on our filing cabinet is increased to 1100 N, so that the cabinet starts to slide. If its acceleration is 7 m/s2, what is the coefficient of kinetic friction between the floor and the cabinet?

N. J. Smith, MTSU, June 2009

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Example 3

Solution

Why is it better not to skid when you want to stop a car in as short a distance as possible?

For example, suppose that the car has a mass of 1200 kg and is travelling at 22 m/s. The coefficient of static friction between the car tires and the road is 0.8, and the coefficient of kinetic friction between the tires and the road is 0.6. How far does the car need to come to a complete stop if it is (a) skidding? (b) not skidding? N. J. Smith, MTSU, June 2009

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What causes friction?

Surface roughness? Will two smooth surfaces slide over each other more easily than two rough surfaces? Cold welding

N. J. Smith, MTSU, June 2009

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Springs and the spring force Consider a spring which is neither stretched nor compressed. The length of the spring in this state is its equilibrium length. We can compress or stretch the spring. Let d be the distance between the equilibrium length and the position of the end of the spring when it is stretched or compressed. d d

Experimentally, it is found that for many materials, the magnitude of the restoring force, or spring force, is proportional to the distance d:

This is not a vector equation. The constant of proportionality, k, is a measure of how stiff the spring is.

N. J. Smith, MTSU, June 2009

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Hooke’s Law Hooke’s law is not really a law. It would be more correct to call it “Hooke’s rule of thumb.” Nevertheless, it is amazingly useful, and many materials obey Hooke’s law (under the right conditions). Hooke’s law relates the magnitude of the restoring force of a spring, to the distance from equilibrium of the spring. A note: You will often see Hooke’s law given as

This is sloppy notation, and can lead to confusion. N. J. Smith, MTSU, June 2009

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Example 4: A typical spring problem A spring with a spring constant of 16.7 N/m is suspended from the ceiling. With no mass attached, it has a length of 12.3 cm. When a mass is hung from the spring, its length increases to 15.6 cm. How much mass was suspended from the spring?

Solution

N. J. Smith, MTSU, June 2009

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Example 5: Bringing it all together A block, incline, and spring are configured as shown in the diagram. The block has a mass 0.75 kg, the spring constant has the value 107 N/m, and the block is pulled down the incline a distance of 7.2 cm from the spring’s equilibrium position and released. What is the initial acceleration of the block if the coefficient of friction between the block and the surface is 0.25 and the angle of the incline is 40°? Solution

N. J. Smith, MTSU, June 2009

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The bigger picture Newton’s second law is amazingly useful: Gets us to the moon. Calculate the position of the planets. Ballistics.

For such a simple equation, it can be very difficult to solve...

N. J. Smith, MTSU, June 2009

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A closer look at Newton’s second law Newton originally stated his law in a different way:

It is a differential equation.

Example: Objects orbiting each other due to the gravitational force.

N. J. Smith, MTSU, June 2009

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The bigger picture cont... Hooke’s law can be used to model the interaction of atoms in a solid.

Atoms are continually jiggling about their equilibrium position. As long as the jiggles are small, Hooke’s law can be used to model their vibrations.

N. J. Smith, MTSU, June 2009

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