You are a medical consultant to a health technology company evaluating an inexpensive traction system for exerting a predictable force at the location of a patient’s leg injury. The device consists of a foot-strap connected to a weight by a rope that goes over a pulley. You are worried that the force exerted on the injury will change when the angle of the leg changes. As a first step in understanding the situation, you decide to model the portion of the patient’s leg below the injury with a cart on an inclined track. The traction device is a simple mechanism for putting steady tension on a leg held at an angle through the use of a hanging weight. You model the system using a ramp, spring, cart, and hanging weight. The spring is attached to the cart and the end of the ramp; by measuring the extension of the spring you can determine the force on the cart. A string is attached to the other end of the cart which goes over a pulley at the end of the track where it is attached to a hanging object. You have been asked to calculate the forces acting on the cart, and how they may depend on the angle of the ramp. If you correctly understand the force balance you will be able to predict the spring extension in your model. You will test your calculations in the lab by building the model and exploring different weight and angle combinations.
Question: 1. Draw a picture of the cart, spring, hanging weight and inclined track arranged as in the figur...
1. Draw a picture of the cart, spring, hanging weight and inclined track arranged as in the figure. Draw a convenient coordinate system to serve as a reference for the orientation of forces.
2. Draw and label the forces acting on the cart. Draw and label the forces acting on the counterweight. How does the force exerted on the string by the hanging object relate to the force exerted on the cart by the string?
3. Draw a free-body diagram of the cart. Is the cart in equilibrium? Use a convenient coordinate system so that you can easily find the components of the forces.
4. Write down the equations that give the condition of equilibrium of the cart and the counterweight. Remember to write a separate equation for force components along each axis of your coordinate system.
5. Calculate the magnitude of the force exerted by the spring on the cart, as a function of the mass of the hanging object, the mass of the cart, and the angle of the track. For data analysis, it will be useful to calculate the amount the spring stretches, rather than the force exerted on the spring, so determine this as well.
6. Sketch a graph of spring stretch vs. track angle, for constant masses. Does the stretch ever equal zero? If so, under what conditions?
7. Beginning with basic physics principles, write down an equation that gives the force exerted by the spring on the cart in terms of quantities you can measure. Make sure that you state any approximations or assumptions that you are making. What quantities must you calculate? What quantities will you measure? Which ones might you adjust and which will remain constant? How does the spring force relate to the appropriate masses and the track angle?
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