6.1 - Circular Motion

Rotational Motion

In addition to translational motion, objects that are not points, and systems of objects, can rotate. We shall consider only rigid objects (rigid bodies) which have a fixed shape. For now, the rotation is around a single line in space called the axis of rotation.

Polar Coordinates

Every point in the object moves in a circle. Point P has coordinates x,y,z and is a distance r from the axis. At time t the position vector makes an angle θ with the x axis.

Because the motion is circular the coordinates x and y are not very useful. The more useful coordinates are the polar coordinates r and θ.

Angular Velocity

• At time t_1 the angle is θ1; at time t _2 the angle is θ2.

The average angular velocity ω is defined to be: 

ω¯= change in angle/change in time = (θ2−θ1)/(t 2−t 1) = Δθ/Δ t

The units of ω are radians/s (rad/s). By convention, if ω is positive the rotation is counterclockwise (CCW), if it is negative the rotation is clockwise (CW).

• When the time interval approaches zero the average angular velocity becomes the instantaneous angular velocity ω.

Angular Acceleration

At time t_1 the angular velocity is ω1; at time t_2, the angular velocity is ω2.

The average angular acceleration α is defined to be

• α¯= change in angular velocity / change in time = (ω2−ω1)/ (t 2−t 1) = Δ ω / Δt

The units of α are rad/s^2.

When the time interval approaches zero the average angular acceleration becomes the instantaneous angular acceleration α.Any point in the object still has a linear (tangential) velocity, speed and acceleration.

• The velocity vector always points along the tangent to the circle.

If the distance of the point is r from the axis then the linear speed is

• v= r × ω

In a rigid object the tangential speed increases with distance from the axis of rotation.

• The circumference of the circle grows but the period does not.

An object moving in a circle is accelerating. The acceleration is related to the angular acceleration and angular velocity. The general relation is complicated because in general there are two components of the acceleration.

• One points towards the center and is called the centripetal acceleration a_C

• The centripetal acceleration causes the velocity to change direction only.

• The other points in the same direction as the velocity (or opposite it) and is called the tangential acceleration, a_tan.

• The tangential acceleration changes the size of the velocity (speed) only.

They are related to ω and α by the equations.

• a_C=r×ω^2 = v^2/r

• a_tan =r×α

In the special case where a_tan is zero, the acceleration points towards the center of the circle, and the object’s speed is constant.

• This kind of motion is called uniform circular motion.

  

• In general the acceleration points in a non-central direction. 

  • This kind of motion is called non-uniform circular motion.
    

Centripetal Force

• The centripetal acceleration occurs due to the application of a force called the centripetal force.

  • Typically the force is a tension in a string, gravity, or a normal force, it is not some new force in the problem.

• Since we know the centripetal acceleration, the force must be:

  • ∑ _radial components F= FC=maC= mv^2 / r = m × r × ω^2

• The sum is over the radial components of the applied forces.

  • If an applied force has no radial component then it doesn't contribute.

  • If the component points towards the axis of rotation it is positive, away from the axis of rotation it is negative.

A rider on a Ferris wheel moves in a vertical circle of radius r at constant tangential speed v.

• How does the normal force that the seat exerts on the rider change compared to the rider’s weight at the top and bottom of the wheel?

  • Another common instance of circular motion occurs when a car rounds a curved road.

• The centripetal force – the force that causes the car to follow the curved road – is friction and/or the normal force if the road is banked.

  • If the wheels are not skidding then the point on the tire in contact with the ground is not moving plus the friction is perpendicular to the motion so the appropriate friction to use is static friction, not kinetic or rolling.

Centrifugal Force

In the frame of reference of an object moving in a circle, there is a force pushing on the object in the outward direction. This force is called the centrifugal force.

• Centrifugal force, often referred to as a "fictitious" or "pseudo" force, is a concept in physics that arises in a rotating or non-inertial frame of reference. It appears to act outward from the center of rotation, opposing the centripetal force, which is directed toward the center of rotation to keep an object in circular motion. Here are some key notes about centrifugal force:

  • Fictitious Force:

    • Centrifugal force is not a real force like gravity or electromagnetism. 

      • It is a perceived force that appears to push objects away from the centre of rotation when you are observing the motion from a rotating frame of reference.

    • It is a consequence of inertia and the tendency of objects to move in a straight line, rather than following a curved path.

  • Centripetal Force:

    • In a rotating system, such as a spinning object or a car moving in a circular path, there must be a centripetal force acting on the object to keep it in its circular path. 

      • This centripetal force is directed toward the centre of rotation.

    • The centrifugal force is often introduced as an apparent force that seems to counteract the centripetal force. 

      • In reality, the object is simply moving in a straight line, but because of its inertia, it appears to be pushed outward.

  • Magnitude of Centrifugal Force:

    • The magnitude of the centrifugal force is directly proportional to the square of the angular velocity (rate of rotation) of the system and the distance of the object from the center of rotation.

    • Mathematically, the centrifugal force can be calculated as F_c = m (ω^2) r, where F_c is the centrifugal force, m is the mass of the object, ω is the angular velocity, and r is the distance from the center of rotation.

• Examples:

  • Common examples of centrifugal force include 

    • the sensation of being pushed outward when turning a curve in a car

    • the clothes sticking to the outer drum of a washing machine during the spin cycle

    • objects moving away from the center of a merry-go-round when it spins.

• Non-Inertial Frames of Reference:

  • Centrifugal force is most often discussed in the context of non-inertial frames of reference. 

    • In an inertial frame (one at rest or moving at a constant velocity), the concept of centrifugal force is not necessary, as objects follow a straight-line path unless acted upon by a real force.

• Understanding Circular Motion:

  • It is important to recognize that centrifugal force is a useful concept for understanding motion from a rotating frame of reference, but it does not represent a real physical force. 

  • The actual force responsible for keeping an object in circular motion is the centripetal force.

6.2 - Newton’s Law of Gravitation

Newtonian Synthesis

• Until Isaac Newton, it was thought that the ‘physics’ of things outside the Earth – the celestial - was different from that of things on the Earth – the terrestrial. Newton rejected this division and applied the same physical laws to both.

• The force that causes an apple to accelerate as it falls is the same force that holds the Moon to the Earth.

• This unification of the physics governing the celestial and terrestrial is known as Newtonian Synthesis. 

  • Newtonian Synthesis had such a profound effect that one of its consequences was the American Revolution.

• Newton tried to find one ‘Law of Gravity’ to explain the motion of objects here on Earth and the motion of the planets. He was aided by 3 facts:

• He knew the force of gravity upon objects here on Earth. If ‘1’ is the object and ‘2’ is the Earth then the weight of object 1 due to object 2 is

  • F_on1from2 = m_1×g_2 where g2 is the acceleration due to the gravity from object 2)

• His 3rd Law tells him that this must also be the force on Earth from the object

• F_on2from1=m_2×g_1 where g1 is the acceleration due to the gravity of the object.

• The motion of the planets around the Sun is described by three laws found sometime earlier by Kepler.

Kepler’s Laws

Many decades before Newton, Johannes Kepler discovered three laws of planetary motion based on observation. 

• The path of each planet about the Sun is an ellipse with the Sun at one focus.

• An imaginary line drawn from each planet to the Sun sweeps out equal areas in equal times.

• The square of a planet’s orbital period is proportional to the cube of its mean distance from the Sun.

• Since the Sun is at a focus, the mean distance from the Sun is the same as the semimajor axis.

• T^2∝s^3

• The ratio of s3 / T2 is the same for all planets.

  • s_1^3 / T_1^2 = s_2^3 / T_2^2

Newton’s Law of Universal Gravitation

• The first two ‘facts’ tell Newton the force of gravity between two objects must be proportional to the product of their masses.

  • g1 must be proportional to m1, g2 must be proportional to m2

• What he doesn’t know is if it depends upon anything else. Newton tried an idea that others had suggested: the force of gravity decreases as the square of the distance between the two objects.

  • This is called an inverse square law.

• With this assumption and his three Laws of Motion, he is able to derive Kepler’s three laws.

  • Later he shows that any other scaling with distance does not give Kepler’s three laws.

• Newton’s Law of Universal Gravitation can be stated as:

Every particle in the universe attracts every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. This force acts along a line joining the two particles

• In mathematical form:

• G is a constant called Newton's Gravitational Constant, or Big G, and has a  value of G = 6.67 x 10^-11 N m^2 / kg^2

• Gravity is quite different from the other forces we have met: it acts even though two objects are not in contact.

  • This is sometimes called Action At A Distance.

  • Something must happen to change the space between the two objects.

Gravitational Fields

• "Gravitational field notes" likely refer to observations, measurements, or data collected regarding the gravitational field in a particular location or region. 

  • The gravitational field is a region in which an object with mass experiences a force due to gravity. Here are some key points that might be included in such notes:

• Location and Coordinates:

  • The specific location where the observations or measurements were taken, including latitude, longitude, and elevation.

• Gravitational Field Strength (Intensity):

  • Measurements of the strength of the gravitational field. 

  • This is typically expressed in units like newtons per kilogram (N/kg) or as the acceleration due to gravity (9.81 m/s² on the surface of the Earth).

• Variations in Field Strength:

  • Notes on any variations or anomalies in the gravitational field strength within the area of study. 

  • These could be due to geological features, underground structures, or other factors.

• Instrumentation and Methodology:

  • Details about the equipment or instruments used to measure the gravitational field. 

  • This might include gravimeters, accelerometers, or other specialized devices.

• Temporal Variations:

  • Any observations or measurements related to how the gravitational field strength changes over time. 

  • This could be due to tides, seasonal variations, or other factors.

• Comparisons with Known Data:

  • Comparisons of the collected data with existing records or models of the gravitational field for that region. 

  • This helps to validate the measurements and identify any discrepancies.

• Potential Applications:

  • Speculation or discussion on how the observed gravitational field might impact various applications. 

    • For example, it could be relevant in geophysics, navigation, or even space exploration.

• Other Pertinent Observations:

  • Any additional relevant observations that might affect or be affected by the local gravitational field. 

    • This could include phenomena like subsidence, seismic activity, or magnetic anomalies.

• These field notes would be valuable for researchers, geophysicists, engineers, or anyone working on projects where an understanding of the local gravitational field is important. 

• They serve as a record of the conditions and data collected, which can be referred to for analysis and comparison in the future.

  • If the mass is m then the force of gravity on that object is mg.

  • g = F / m, or Gravitational Field Strength (g) is equal to Gravitational Force (F) over Mass of the Object (m)

Force fields are often represented by field lines.