Principles of Technology Prime Movers in Mechanical Systems Introduction Force and torque are the two prime movers in any mechanical system. Force is the name given to a push or pull on an object that tends to want to cause the object to move along a straight path, while torque is name given to a push or pull that tends to want to make the object rotate around a fixed point. Torque is often referred to as the rotational counterpart to force. Clearly, it s possible to push on an object and not have it move perceptibly. In this case we say that the object doesn t move, but it changes shape, or deforms. In fact, when an object deforms it does, in a sense, move, but not in the way we commonly think of movement or motion. If an object doesn t move when forces act on it, we say the object is in equilibrium and that the forces are balanced. Another way of saying the same thing is that the net force acting on the object is zero. Look at the computer monitor that s likely sitting in front of you as you read this. The force due to gravity (weight) is pushing down on the monitor. This force is balanced by the force that the table is applying upward (normal force) on the monitor. The forces are balanced, the monitor is in equilibrium, and the net force is zero. A different, but related idea is what is often referred to as action reaction. The force the monitor is applying to the table is equal and opposite (in direction) to the force that the table is applying to the monitor. This will always be true regardless of whether the objects in question are moving or not. This is Newton s Third Law of Motion, which states: If object A exerts a force on object B, then object B exerts a force of equal magnitude back onto object A but in the opposite direction. This same discussion also applies to torque. Think of the last time you used a wrench to loosen a nut. You start by applying a perpendicular force on the end of a wrench. If the nut does not move, the net torque is zero, or you could say the torques are balanced, or you could say that the wrench (or nut) is in equilibrium. Said another way, the torque you appliy to the wrench is equal and opposite to the torque that friction is applying (in the opposite rotational direction) on the wrench. Once the force of friction is overcome, the nut starts to turn and the net torque is no longer zero. Oh, and you bark your knuckles on a protruding piece of metal that s always fun. 1
It s important to note, that even if the nut is moving, the force that you re applying to the end of the wrench is equal and opposite to the force the wrench is applying to your hand. So Newton s Third Law holds whether the torques (or forces) are balanced or not. This may seem contradictory and/or confusing. To clarify: Newton s Third Law deals with two objects acting on each other. The equilibrium concept deals with forces acting on a single object. Again, Newton s Third Law holds regardless of whether or not an object is in equilibrium. To calculate the net force acting on an object, simply add up all of the forces. F... NET F1 F2 F3 F N If all of the forces act along the same line (such as in the case of a tug of war) then put a negative sign in front of either the forces pulling to the left or the forces pulling to the right. It doesn t matter which. Then add the all of the forces together, both positive and negative. If the net force is zero, then there is no motion to the left or to the right. If the net force is not zero, then somebody is headed for the mudhole. Calculating the net force acting on an object when the forces don t act along a line is more complicated and requires trigonometry or a graphical method. Calculating torques is more complicated for two reasons. First, it depends on what is meant by calculate the torque. If the goal is to calculate the net torque, then the method is analogous with the one above regarding forces, except that instead of pulls to the left and right, a discussion of clockwise and counter-clockwise twisting becomes relevant.... NET 1 2 3 N Note that in the equation above the Greek letter tau τ is used to symbolize torque. However, there is another meaning to calculating torque. Recall the earlier discussion about using a wrench to loosen a nut. If the nut won t budge, what can be done? There are several answers, but let s focus on two simple ways of increasing the torque applied to the nut by the 2
wrench. First, the person pushing on the end of the wrench could push harder. Second, the person could get a longer wrench. Third, the person could do both! L With respect to applied force and wrench length, the torque can be calculated as F L where τ is the torque, F is the applied force and L is the length of the wrench. It is important to note that this formula assumes that the applied force is perpendicular (at right angles) to the wrench length. If not, the formula becomes F L sin where is the smaller angle between the wrench handle and the applied force. This idea can be applied to other situations, including pulleys and gears. In that case the radius of the gear replaces the length of the wrench. Also the (force) angle between the gears is always 90, so the more complicated equation (above) reverts to the simpler form. Because of the dependency of torque on force it is possible to gain a mechanical advantage when a small gear drives a larger one. The forces between the teeth of the gears is equal and opposite (Newton s Third Law), but the torque produced by the larger gear will be greater due to its larger radius. Finally, from your experience you may already know the units for torque. However, take another look at the equation above. Force times length equals torque. The units for force are the pound or the newton, while the units for length are the foot and the meter. This means that the fundamental units for torque are the lb ft or the N m. F L lb ft or F L N m It is important that the unit for force is written in front of the unit of length so as not to confuse the measurement of torque with the measurement of energy which, as you will later is derived from the same base units. F 3
HW 4: Torque Principles of Tech 1 DUE Monday, 5/08/17 Be sure to review and follow the HOMEWORK GRADING CRITERIA prior to submitting this assignment for grading. 1. If a force of 56.2 pounds is applied to a torque wrench 1.50 ft. long, what torque in lb ft is indicated by the wrench? 2. A torque of 175 lb ft is needed to free a large rusted-on nut. The length of the wrench is 1.10 ft. What force in pounds must be applied to free it? 3. A motorcycle head bolt is torqued to 30.0 N m. What length shaft is needed to exert a maximum force of 80.0 N? 4. A force of 112 N is applied to a shaft of radius 3.50 cm. What is the torque on the shaft in N m? 5. Mike s car has a flat tire. He is trying to remove a lug nut that the tire shop over-tightened with an impact wrench. Mike applies a force of 80 pounds to the end of a lug wrench that is 14 inches long. a. Find the torque (lb ft) applied to the lug nut by the wrench. b. The torque Mike applies to the lug nut does not make it turn. What is the net torque on the lug nut? Explain. c. Describe any torques, other than Mike s, acting on the lug nut. 4
6. Sandra and Maria are loading their ski boat onto a trailer. The trailer has a hand winch to pull the boat into position on the trailer. The winch has a pair of gears to multiply the torque Sandra produces when she pulls or pushes on the winch handle. One gear is 4 inches in diameter and the other is 11 inches in diameter. A torque of 10 lb ft is applied to the smaller gear. What torque (lb ft) is produced by the larger gear? 7. Maria has severe arthritis and can only apply a maximum force of 25.5 N when opening her front door. (a) How much more torque would Maria be able to apply if she purchased a lever style door opener with a handle 12.7 cm long compared to her conventional doorknob with a radius of 3.74 cm? (b) How much less force would she need to apply to the new door handle in order to maintain the original amount of torque she had with her old doorknob? 8. Using two gears, how can you double the torque of a motor? Explain clearly and completely and/or give an example using numbers. If you give an example using numbers, be sure to show your work. 9. You are installing a new spark plug in your car, and the manual specifies that it must be torqued to 45N m. Using the data in the drawing to the right, determine the magnitude of the force that you must exert on the wrench 10. Find the net torque (magnitude and direction) produced by the forces F 1 and F 2 about the rotational axis shown in the drawing to the left. Neglect the mass of the rod and assume it does not flex. 5