Physics 141Mechanics Lecture 5 Reference Frames With or without conscience, we always choose a reference frame, and describe motion with respect to the reference frame, onto which we attach our coordinate system. Ordinarily, we use the ground as the reference frame. Sometimes other frames are necessary, for example if you take a journey in a cruiser in the ocean. Unable to see the ground, you’d use the ship as the reference frame. We need to formulate the relation between the same motion with respect to different reference frames. In this course, we only consider frames that move at constant velocity. Such frames are called inertial reference frames.

Relative Motion of Galilean Transformation y x Frame A Frame B x y r BA r PA r PB Assume that frame B is moving at constant velocity V BA with respect to frame A and the two frames coincide at t=0. The origin of frame B at time t is at position r BA with respect to frame A, r BA =r 0 +V BA t. Galilean transformation is based on the assumption that the time is the same in both frames The length is also assumed to be the same. Then V BA

Example: The rain drops are falling with speed v rg with an angle of  eastward from vertical. If you are driving with speed v cg to the north, what will be the velocity of the rain drops and their angle from vertical that you can see? Solutions: Rain velocity to the ground The car velocity to the ground v cg =v cg j From Galilean transformation The rain drop’s velocity to the car Angle from vertical

The Relativity of Time The assumption in Galilean transformation that time and length are the same in different frames is strictly speaking inaccurate. Einstein's postulates: –Physics laws are the same in all inertial frames. –The speed of light is the same in all these frames. Time dilation. If S' moves at V with respect to S, and light travels in S' perpendicular to V for a distance h and bounces back. In S',  t'=2h/c. In S, So, a moving clock is observed to run slow.

Length Contraction The measured distance between two events depends on the frame relative to which it is measured. Suppose S' is a train moving at V with respect to the station S in which a light pole Q serves as a marker. To some one in S, the length of the train is its speed times the time interval to pass the pole l=V  t. To some one in S', the length of the train is l 0 =V  t'. Now in S the events happen at the same location, the time-dilation formula tells us  t'=  t. So, a moving object is observed to be contracted.

Transformation in Special Relativity More accurate transformation is the Lorentz transformation, for V along x-axis where When v A is parallel to V and along x-axis, which goes to the result of Galilean transformation if either V/c<<1 or v B /c<<1

Newton’s First Law Newton’s first law states that a body at rest remains at rest and a body in motion continues to move at constant velocity unless acted upon by an external force. The property of any object keeping its motion unless disturbed by others is called inertia. Newton’s first law is also termed as the law of inertia. The law is valid in an inertial frame, which does not accelerate with respect to the rest of the Universe. Newton’s thought was: if an object moves on a level floor, it stops because of friction. If the floor is smooth, the object goes straight further. In the case of no friction, it will move unchanged forever.

Newton’s Second Law Newton’s second law of motion is which states that the change of motion, the acceleration a, of an object of mass m, is proportional to the force F acting on the mass and inversely proportional to the mass. Note it is a vector equation. The unit of force in SI is Newton (N) 1 N = 1 kg m/s 2 If more than one force is acting on the object, we have to find the vector sum of all the forces first:

Newton’s Third Law Newton’s third law states that whenever an object exerts a force on another object, the latter exerts a reaction force of equal magnitude and opposite direction on the former: The action and reaction forces are acting on different bodies, so they cannot cancel each other. A F BA B F AB

Example: Assuming a force F is acting on two bodies m and M. Find a. Solution: Taking m and M as one object of mass m+M, then the acceleration is a=F/(m+M) What is the force between the bodies? For M the force by m gives its acceleration, so From the Newton’s third law, the reaction force on m is F Mm =-F mM. For m the total force is It can be seen that m M F

More about Force Force is a vector. If more than one force is acting on a body, we first have to find the vector sum of all the forces on the body in order to apply Newton’s second law and find out the acceleration. There are three important elements for any given force: magnitude, direction, and point of action. The significance of the third element will be apparent when we study rotational motion and torque.

Common Forces The weight of a body W is the force on the body by the gravitational attraction of the Earth. The normal force N is the force by the supporting surface on the body and is perpendicular to the surface and pointing toward the body. Friction f is another common force occurring on any contacting surface. The direction of the force is parallel to the surface, and pointing always against the intended direction of motion. Tension T is the force exerted on a body by an attached string, rope, cable, etc. Its direction is always away from the body and along the string, and is acting at the point of attachment. The magnitude of the tension is the same throughout the string, if the mass or the acceleration of the string is negligible.