Springs. A spring exerts a force when stretched or compressed that is proportional the the displacement from the uncompressed position: F = -k x

Transcription

1 Springs A spring exerts a force when stretched or compressed that is proportional the the displacement from the uncompressed position: F = -k x where x is the displacement from the uncompressed position and k is called Hooke s constant. Hooke s constant has units of N/m. The minus sign indicates the force always points in the direction opposite to the displacement. The scales in a produce section of the market are simply a spring that moves a pointer to indicate how much gravitational force is being placed on the basket e.g., the weight of the apples you are purchasing. The bigger the force (larger weight of apples) stretches the spring more and moves the pointer more. Springs come in all shapes and sizes. The shock absorbers on your car are normally surrounded by a very large spring. Mechanical clocks and watches used springs to store energy to run the watch or clock. Even your bed (the box springs ) have springs to make your bed more comfortable. Springs do not have to be a coil of metal. Many things work like springs depending on the shape of the object. The spring in a mechanical watch is a spiral coil of metal. An archer s bow works

2 like a spring. The further back you pull the bow, the larger force the bow applies to the arrow. A steel bridge span sags (slightlywe hope) under the force of a truck. The force of this spring bridge balances the weight of the truck. Translational Equilibrium When the net forceonanobject iszero, there isno acceleration. We say the object is in equilibrium. If the object is at rest, then it stays at rest. Consider standing on two bathroom scales. If you distribute your weight evenly between the two scales, each scale reads half of your weight. If you lean to one side, one scale reads more than the other but the sum of the two scales is your weight. In each case, the sum of the normal forces (reaction force) of the scales just balances your weight. The forces don t have to be two equal and opposite forces. If you stand on two bathroom scales, your weight is the sum of the two readings on the scales. As you shift your weight slightly from one scale to another, the number on an individual scale will change but the sum is always equal to your weight. The forces don t even have to be in the same direction. Consider a sign hanging from two wires: wire 45 o sign 45 o wire ceiling weight of sign

3 The sign does not move i.e., it is in equilibrium, but the forces of the wires and gravity are all in different directions. Rotational Equilibrium Things can be in rotational equilibrium as well as translational equilibrium. Both of the seesaws in the figure are in rotational equilibrium. For the left case, the net torque is zero. Each side of the horizontal seesaw has a force of F and a lever arm of L. However, the left force, F, produces a positive torque (τ = F L) (counterclockwise rotation) and the right force, F, produces a negative torque (clockwise rotation τ = -F L). The magnitudes of the torques are equal but one is positive and one is negative so they cancel out. For the right case the net torque is also zero. The force F has a lever arm of L. The torque is positive since it want to rotate the seesaw counterclockwise. The force on the right (2F) has a lever arm ofl/2soitalsoproducesatorquewithamagnitudeoff L.However, it trys to produce a clockwise rotation which makes it negative and it cancels out the torque on the left for a net torque of zero. L L L 1/2L F F F 2F A final comment about equilibrium, Things can be moving and be in translational or rotational equilibrium. To be in equilibrium, the net force or net torque must be zero. Something can be moving with constant velocity or constant rotational speed and still be in equilibrium. Say youareslidingacrateacrossthefloorwithconstant velocity. Since the velocity is constant, the is no acceleration. You

4 are pushing with some force but there is also friction between the floor and the crate. Your pushing force is just balanced by the force of friction if the crate has constant velocity. v = constant You push crate friction Bouncing Balls Balls are object that behave as spherical springs. If you hold a ball between your finger and thumb and squeeze, it applies a force back. Squeeze it harder and it pushed back harder. Like springs, balls are different. Some bound better than others just like some springs are stiffer than other springs. The traditional way to measure the bounce of a ball is the coefficient of restitution: coefficient of restitution = outgoing speed of ball incoming speed of ball This ratio can never be greater than 1. Another measure of how well the ball bounces is the ratio of the outgoing(rebound) energy to the incoming (collision) energy. This ratio must also be less than 1. The two are clearly related because kinetic energy is proportional to the square of the speed. If we look at the table in the book:

5 Rebound Energy Type Collision Energy Coef. of Restitution Superball Racquet ball Golf ball Tennis ball Steel ball bearing Baseball Foam rubber ball Unhappy ball Beanbag wecanseethisrelationship. Noticethattheenergyratioisthesquare of the coefficient of restitution in all cases. Of course, the hardness of the surface also determines how well the ball bounces. If you drop a golf ball on a floor it bounces better than on a mattress. (The values in the Table are from bouncing from a hard surface.) The surface on which the ball bounces can also acts like a spring. A good tennis racket also deforms (storing energy) as well as the ball. What is the source of the extra speed of the ball when hit with a good racket? A person on a trampoline is a more extreme case of a lively surface. You can take a beanbag (very low coefficient of restitution) and hititwithatennisracket orbaseballbat andgiveitalotofoutgoing speed. In this case the surface is moving even if the object has a low

6 incoming speed. We need to re-define our definition to account for a moving surface like a bat: speed of separation coefficient of restitution = speed of approach. Ifthebatiscomingatthebaseballat20m/sandtheballismoving at 30 m/s, the relative motion between the ball and the bat makes the collision like a collision of a 50 m/s baseball with a stationary bat (or a 50 m/s bat hitting a stationary baseball). There is more than just the collision speed involved in the collisions between the bat and the ball. If the ball hits at a place that is offcenter of the center of mass of the bat, the bat will experience not only the force of the collision but a torque. The same is true of the ball. Iftheballcontactsthebataboveorbelowtheitscenterofmass, the ball will experience a torque and be set in rotational motion as well as the desired linear motion. Bats are not perfectly rigid objects. All real objects have vibrational modes. These can be very complex patterns. Engineers designing bridges have to pay attention to this vibrational modes. For a baseball bat, the design is such that the center of mass is located near an node where the bat gives the ball the best distance. In effect, the ball does not waste energy vibrating the bat.

7 Uniform Circular Motion Riding on a Merry-go-Round (Carousel), riding in an automobile making a turn on a road or a satellite moving around the earth are all examples of uniform circular motion. Uniform circular motion is defined as: Uniform circular motion is the motion of an object traveling at a constant (uniform) speed in a circular path. v a r One of the first things we notice about an object going around a circle with constant speed is the time it takes to make one complete trip around the circle. This is called the period. If r is the radius of the circle, the distance around the circle is 2πr. the speed, v, is so the period is: v = 2πr T T = 2πr v By the definition, the speed is constant in uniform circular motion. What about the velocity? The velocity is tangent to the circle of motion so the direction changes as we go around the circle. Thus the velocity changes. Since the velocity is changing, there must be an acceleration. This acceleration is given by: acceleration circular = v2 r

8 What about the direction of the acceleration vector. The acceleration is the change in the velocity. It has to continuously change the velocity vector back towards the center of the circle to keep the object moving in a circle. Forces on a Roller Coaster A roller coaster is a good example of how work, energy, gravity and centripetal force can interact. At the start of the roller coaster ride, a chain drive pulls the car to the top of the first (and normally tallest) hill. Work is done on the car by the chain drive and the car gains some kinetic energy and a lot of gravitational potential energy. The gravitational potential energy is converted to kinetic energy as your car rolls down the hill. Of course, some of the energy is dissipated as heat by friction. At the bottom you encounter a loop the loop section of the track. Nowcentripetalforcecomesintoplay. Thetracksforcethecarintoa (vertical)circle. Asthecarenterstheloop,itistravelingalongwithit kinetic energy at constant velocity. Half way up the loop, you still feel gravity pulling you down. The car is slowing down (kinetic energy is being converted to gravitational energy). There is also the centripetal force on the car (and you in the car) because the track is pushing the car in the vertical circle. You feel and acceleration towards the

9 center of the circle. The total force on the car is the vector sum of the downward force of gravity and the horizontal(inward) centripetal force. When you areat the top of the loop, gravityis still downward. If you car has just enough kinetic energy to get it over the top of the loop, gravity is now providing all the centripetal force the car needs. The car (and you) would be weightless at the top. Notice when you are at the top, gravity is the same as always but you feel nearly weightless i.e. you are in free fall (or near to free fall). This apparent weight is due the car s (and your) inertia. The car has to have the force from the track to keep it moving in a circle. You want to go straight because of your inertia. You feel an acceleration outward like when you are pushed to the outside of a turn when traveling in a car.