Applications of Newton’s Laws. Strings and Springs When you pull on a string or rope, it becomes taut. We say that there is tension in the string.

Applications of Newton’s Laws

Strings and Springs

When you pull on a string or rope, it becomes taut. We say that there is tension in the string.

Strings and Springs

The tension in a real rope will vary along its length, due to the weight of the rope.

Here, we will assume that all ropes, strings, wires, etc. are massless unless otherwise stated.

Strings and Springs

An ideal pulley is one that simply changes the direction of the tension:

Strings and Springs

Hooke’s law for springs states that the force increases with the amount the spring is stretched or compressed:

The constant k is called the spring constant.

Translational Equilibrium

When an object is in translational equilibrium, the net force on it is zero:

(6-5)

This allows the calculation of unknown forces.

Translational Equilibrium

Connected Objects

When forces are exerted on connected objects, their accelerations are the same.

If there are two objects connected by a string, and we know the force and the masses, we can find the acceleration and the tension:

Connected Objects

We treat each box as a separate system:

Connected Objects

If there is a pulley, it is easiest to have the coordinate system follow the string:

Frictional Forces

Friction has its basis in surfaces that are not completely smooth:

Frictional Forces

The kinetic frictional force is also independent of the relative speed of the surfaces, and of their area of contact.

Frictional Forces

Kinetic friction: the friction experienced by surfaces sliding against one another

The kinetic frictional force depends on the normal force:

(6-1)

The constant is called the coefficient of kinetic friction.

The static frictional force keeps an object from starting to move when a force is applied. The static frictional force has a maximum value, but may take on any value from zero to the maximum,

Frictional Forces

depending on what is needed to keep the sum of forces zero.

Frictional Forces

(6-2)

where

(6-3)

The static frictional force is also independent of the area of contact and the relative speed of the surfaces.

Frictional Forces

Newton’s Law of Universal Gravitation

Newton’s insight:

The force accelerating an apple downward is the same force that keeps the Moon in its orbit.

Hence, Universal Gravitation.

Newton’s Law of Universal Gravitation

The gravitational force is always attractive, and points along the line connecting the two masses:

The two forces shown are an action-reaction pair.

Newton’s Law of Universal Gravitation

G is a very small number; this means that the force of gravity is negligible unless there is a very large mass involved (such as the Earth).

If an object is being acted upon by several different gravitational forces, the net force on it is the vector sum of the individual forces.

This is called the principle of superposition.

Gravitational Attraction of Spherical Bodies

Gravitational force between a point mass and a sphere: the force is the same as if all the mass of the sphere were concentrated at its center.

Gravitational Attraction of Spherical Bodies

What about the gravitational force on objects at the surface of the Earth? The center of the Earth is one Earth radius away, so this is the distance we use:

Therefore,

Gravitational Attraction of Spherical Bodies

The acceleration of gravity decreases slowly with altitude:

Gravitational Attraction of Spherical Bodies

Once the altitude becomes comparable to the radius of the Earth, the decrease in the acceleration of gravity is much larger:

Gravitational Attraction of Spherical Bodies

The Cavendish experiment allows us to measure the universal gravitation constant:

Gravitational Attraction of Spherical Bodies

Even though the gravitational force is very small, the mirror allows measurement of tiny deflections.

Measuring G also allowed the mass of the Earth to be calculated, as the local acceleration of gravity and the radius of the Earth were known.

Circular Motion

An object moving in a circle must have a force acting on it; otherwise it would move in a straight line.

The direction of the force is towards the center of the circle.

Circular Motion

Some algebra gives us the magnitude of the acceleration, and therefore the force, required to keep an object of mass m moving in a circle of radius r.

The magnitude of the force is given by:

(6-15)

Circular Motion

This force may be provided by the tension in a string, the normal force, or friction, among others.

Circular Motion

Circular Motion

An object may be changing its speed as it moves in a circle; in that case, there is a tangential acceleration as well: