Lecture 4 Spring problem and conservation of mechanical energy Hooke's Law The restoring force exerted by the spring is directly proportional to its displacement. The restoring force acts in a direction that would restore the spring to its natural length. Mathematically, F = kx where k is the spring constant which is the measure of how stiff the spring is. The SI unit is N/m. x is the displacement of the end of the spring form its natural position F is the restoring force The negative sign indicates that the restoring force is always in the opposite direction to the displacement. x natural length of the spring the spring is compressed and x is the compression x the spring is extended and x is the extension NOTE Hooke's Law applies to springs within certain limits. If the spring is compressed or extended so much that it is permanently deformed, Hooke's Law no longer applies. The restoring force F and the force that is compressing or extending the spring are a pair of action-reaction forces and they are equal in magnitude and opposite in direction. x is the measure of compression or extension which is the change in length of the spring. The F x graph produces a straight line with gradient of k. restoring force (F) F x extension or compression ( x ) The strain potential energy stored in a spring can be found by calculating the area under F - x graph: 1 1 1 2 Esp = Fx = kx x = kx 2 2 2 Physics 1
F( N) stiff spring weak spring x( m) The gradient of the F x graph is the spring constant k which indicates the stiffness of the material Conservation of mechanical energy Recall that mechanical energy is comprised of kinetic energy, strain potential energy and gravitational potential energy. If the gravitational force and the restoring force are the only forces doing work on an object, the kinetic energy and the potential energy of the object interconvert while the total mechanical energy of the system (comprising the object and the earth or spring) remains constant. Mathematically, Eki + Epi = Ekf + Epf if the weight and the restoring force are the only forces doing work on the object A model rocket of mass 0.20 kg is launched by means of a spring, as shown in figure below. The spring is initially compressed by 20 cm, and the rocket leaves the spring as it reaches its natural length. The force-compression graph of the spring is also shown in figure below. How much energy is stored in the spring when it is compressed? Physics 2
What is the speed of the rocket as it leaves the spring? What is the maximum height, above the spring, reached by the rocket? The air resistance on the way up could be ignored. In a lab class at school, Nick is given a spring with a spring constant of 15 N/m and natural length of 0.30 m. He hangs it vertically, and attaches a mass to it, so that the new length of the spring is 0.50 m. Assuming the spring has no mass, what was the value of the mass he attached? Nick pulls the mass down a further distance of 0.10 m. By how much has the potential energy stored in the spring changed? He now releases the mass, so that the mass-spring system oscillates. Ignore air resistance. Which one of the curves (A-D) below could best represent the variation of the total mechanical energy of the mass-spring system as a function of position? Physics 3
Figure on the left shows an ideal spring with a 2.0 kg mass attached. The spring-mass system is held so that the spring is not extended. The mass is gently lowered and the spring stretches until, in figure on the right, the spring-mass system is at rest. The spring has extended by 0.40 m. What is the value of the spring constant, k, of the spring? What is the difference in the magnitude of the total mechanical energy of the mass-spring system between the two figures? Part of a roller coaster ride at an amusement park is shown in figure above. The car with people in it has a total mass of 800 kg. The cars starts from rest at point A, a vertical height of 20 m above point B. Ignore the effects of friction. Physics 4
What is the speed of the car at point B? A h B The ball starts from rest at point A, a height of h metres above point B. The ball comes to rest momentarily at point B due to friction. If the ball starts at point B with an initial speed of 6.0 m/s down the slope, it will come to a stop at point A. What is the value of h? Elastic or inelastic collision Recall that if two objects collide on a horizontal and smooth surface, then the total momentum of the system comprising the two objects is conserved. But there are differences between two types of collisions. 1. Elastic collision - the total kinetic energy after the collision is the same as it was before the collision. BUT during the collision, some kinetic energy of the two colliding objects is transformed into the strain potential energy due to the temporary deformation of the objects. Since the collision is elastic and the interaction between the two objects can be considered as restoring forces, therefore, the total energy of the system is conserved before, during and after the collision. Total Ek Total energy (kinetic and strain potential) Physics 5
2. Inelastic collision - even though total momentum is still conserved, the total kinetic energy of the colliding objects after the collision is less than it was before the collision due to permanent deformation of the objects. A proportion of the initial kinetic energy is lost in forms of heat and sound. Total Ek Total energy (kinetic and strain potential) Ranjiv, who has a mass of 80 kg, is running with a speed of 4.0 m/s as he steps onto a stationary trolley of mass 50 kg as shown in figure below. Ranjiv holds on to the trolley. Ranjiv and the trolley then move forward together in the same direction. Is this collision between Ranjiv and the trolley elastic or inelastic? Nick and Sarah are studying collision by sliding blocks together on a frictionless table. Nick slides a block of mass 2.0 kg with a speed of 3.0 m/s that collides with a block of mass 1.0 kg, which was at rest. After the collision the 1.0 kg block has a speed of 4.0 m/s. The situations before and after are shown in figure below. Determine whether this is an elastic or inelastic collision. Physics 6