EJC Physics H203 Dynamics 2023 1.Notes (FULL)

Content • Newton’s Law of motion • Linear momentum and its conservation Learning Outcomes Candidates should be able to: (a) State each of Newton’s laws of motion (b) Show an understanding that mass is the property of a body which resists change in motion (inertia) (c) Describe and use the concept of weight as the effect of a gravitational field on a mass (d) Define and use linear momentum as the product of mass and velocity (e) Define and use impulse as the product of force and time of impact (f) Relate resultant force to the rate of change of momentum (g) Recall and solve problems using the relationship F = ma, appreciating that resultant force and acceleration are always in the same direction (h) State the principle of conservation of momentum (i) Apply the principle of conservation of momentum to solve simple problems including inelastic and (perfectly) elastic interactions between two bodies in one dimension (knowledge of the concept of coefficient of restitution is not required) (j) Show an understanding that, for a (perfectly) elastic collision between two bodies, the relative speed of approach is equal to the relative speed of separation (k) Show an understanding that, whilst the momentum of a closed system is always conserved in interactions b etween bodies, some change in kinetic energy usually takes place. In dynamics, we study the forces that act on a body to cause motion . The vector sum of the se forces gives a resultant force that causes the body to accelerate. This resultant force causes change in motion. Dynamics explain the reasons behind k inematics where we describe how a body move s under constant acceleration. In the 17th century, Sir Isaac Newton formulated his 3 laws of motion. The process required Newton to deploy the concept of an “external agent” that can transport action at a distance, provide instantaneous motion and not be subject to resistive forces. Newton’s person al reading habits at that time resulted in him imagining these “external agents ” as angels. He gradually trimmed their wings and transformed this new agent into a purely objective “force”. Today, Newtonian mechanics is useful for many engineering efforts in our everyday scale, like how an artillery shell travels in air, and it describes many phenomena observed. Occasionally, we can take time off to marvel at the inspiration behind “force” – that the wings of angels are ever beating invisibly and constantly providing instant messaging between objects - so that forces may exist in our modern world. The Tesla Model 3 did very well in crash-testing because its engineers understood Dynamics very well in ensuring the safety of the passengers.

What the 3rd law refers to as “a body” can also refer to a collection of bodies. We can regard these bodies as “a system”. Example 1 Newton’s 1 st Law gives rise to the idea inertia: that a body is reluctant to change its “status quo” of motion. True or False: the weight of the ball and the normal contact force on ball by table are Newton’s 3rd law pair of action-reaction forces. Fgrav on ball Fgrav on Earth Non table Non ball Solution False. The weight of the ball is the gravitational force that the Earth acts on the ball. By Newton’s 3rd Law, the reaction force should be the gravitational force that the ball acts on the Earth. For the normal contact force on the ball, the Newton’s 3rd Law pair of action -reaction is the normal contact force on table by the ball. Newton’s Third Law of Motion states that when body A exerts a force on body B, body B exerts on body A a force of the same type, equal in magnitude and opposite in direction. system direction of acceleration Fwall on feet Ffeet on wall Fwall on feet upthrust weight Free Body Diagram of a swimmer in pure horizontal motion. The net force on swimmer is provided by the force of the wall on the feet. Newton’s 3 rd Law applies at the wall: the same type of force (contact force) acts different bodies (the swimmer and the wall) with equal magnitude and opposite directions. Newton’s First Law of Motion states that an object stays at rest or continues to move at constant velocity unless a resultant force acts on it. The mass of a body is the property of a body which resists change in motion.

The weight of a body at a point (location) in space is given by W = mg, where m is the mass of the body and g is the gravitational field strength at that point in space. The mass of a body remains constant anywhere in the universe while the weight changes with gravitational field strength that the body is situated in. Example 2 A fly hovers stationary in front of an open-top rail cart that is at rest. The cart starts to move forward. Explain why the fly will hit the cart. p : linear momentum (kg m s-1) or (N s).) m : mass (kg) v : velocity (m s-1) Linear momentum is a vector quantity and it takes the same direction as the velocity of the body. It takes work done to accelerate a body so that it gains momentum. Conversely, the more momentum a body has, the “harder” it is to reduce the momentum to zero to stop it (see Newton’s 2nd Law). Solution Fly is stationary so is in translational equilibrium with no resultant force. By Newton’s 1st Law, it continues to be at rest. Cart moves from rest so there is change in velocity hence acceleration. Frictional force by track acts on the wheels and cart accelerates towards the fly. The weight of a body is the force acting on the body due to a gravitational field The linear momentum of a body is the product of its mass and its velocity. p = mv

The formal definition of Newton’s 2 nd Law reads mathematically as net d d pFk t= with k denoting the proportionality constant. Considering SI units and regarding 1 N as the force which results in an acceleration of 1 m s -2 when i t is applied to a mass of 1 kg: ( )net d1 d pF t= If the mass is constant, then by product rule ( )net dd dd dd dd pF mv tt vmmv tt ma == =+ = : it reduces to a more familiar form. Example 3 A cricket player catches a fast moving ball with his bare hands. Explain why it is preferable that his palms draw back while catching the ball. Note: (i) netF on ball is in a negative direction i.e. opposing the initial velocity of the ball – in order to slow the ball down. (ii) the body that is having the change in momentum is the body that is experiencing a net force exerted (in this case, the ball). Therefore, it is very common for explanations to demand the action-reaction pair of the force instead (in this case, the hand). Consequently, many explanation-type questions involve both N2L and N3L in a similar fashion. Solution To catch a ball is to reduce the momentum from just before touching the hands, to zero. [N2L] For this same change in ball’s momentum , the time interval when the force is applied by hand on ball to slow it down is lengthened. [force, magnitude] By Newton’s 2 nd Law, the average force on ball by hand is reduced. [N3L] By Newton’s 3rd Law, the average force on hand by ball is reduced. so hand experiences less pain when catching the ball. Newton’s Second Law of Motion states that the rate of change of momentum of a body is [magnitude] directly proportional to the resultant force acting on it and [direction] takes place in the direction of the resultant force. Resultant force is related to the rate of change of momentum. net d d pF t= 0

Example 4 Solution Take direction towards “you” as positive: ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) initial on bouncy final initial on non-bouncy final initial 0 095 20 1 9 1 9 1 9 01 initial momentum Ns force on bouncy ball 38 N0 10 force on non-bouncy ball p mu . .