When a patient’s injured neck is healing, it is often desirable to prevent the weight of the head from pushing down on the neck. This can be accomplished with the system of pulleys shown in the figure. The pulleys are all small and extremely light in weight and have no appreciable friction. Typically, a person’s head makes up 7.00% of his or her body weight.
If the head of a 63.0 kg person is to be supported completely by the apparatus shown, what should the mass of W be? Start with a free-body diagram of the head.
without the diagram of the apparatus, it is not possible.
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Well, it sounds like this patient's neck needs some serious support. Let's see how we can lighten the load, shall we?
First, let's analyze the situation. We have a system of pulleys to prevent the weight of the head from pushing down on the neck. We want to find out what the mass of W should be in order to support the person's head. Easy-peasy, right?
Since we're dealing with a free-body diagram of the head, we need to consider all the forces acting on it. We have the force of gravity pulling the head down, and we want the tension in the rope to counteract that force.
Now, here comes the fun part. We know that the person's head makes up 7.00% of their body weight. So, if we have a 63.0 kg person, the weight of their head would be about 7% of that.
Calculating... 63.0 kg * 0.07 = 4.41 kg. So, the weight of the head is approximately 4.41 kg.
Now, we want to find the mass of W, which is equivalent to the tension in the rope. Since the pulleys are light and frictionless, we can assume that the tension in the rope is equal to the weight of the head.
Therefore, the mass of W should also be 4.41 kg. We want to keep things balanced, after all!
So, there you have it. The mass of W should be 4.41 kg to support the person's head completely. Now, let's hope this apparatus does its job and gives that neck the break it deserves!
To find the mass of W, we first need to set up a free-body diagram of the head and analyze the forces acting on it.
Free-Body Diagram:
- The weight of the head (W_head) acts downward.
- The force exerted by the supporting string (T) acts upward.
- The force exerted by the weight of the person (W_person) acts downward.
Since the weight of the person (W_person) acts downward and we want to prevent the weight of the head (W_head) from pushing down on the neck, the upward force exerted by the supporting string (T) needs to balance out the combined weight of the head and the person.
Now, let's set up the equation based on the principle of equilibrium:
Sum of forces vertically = 0
T - (W_head + W_person) = 0
We know that the weight of the person (W_person) is equal to the person's body weight, which is 63.0 kg multiplied by the acceleration due to gravity (9.8 m/s^2). So, W_person = (63.0 kg)(9.8 m/s^2) = 617.4 N.
Substituting the values into the equation:
T - (0.07 * W_person + W_person) = 0
Simplifying the equation, we get:
T - (1.07 * W_person) = 0
Now, we can solve for the tension in the string (T) by rearranging the equation:
T = 1.07 * W_person
The mass of W can be found using the equation:
W = (mass of W) * (acceleration due to gravity)
Since W is being used to support the head completely, its weight should be equal to the weight of the head (W_head).
W_head = (mass of W) * (acceleration due to gravity)
Substituting the values, we get:
W_head = (mass of W) * (9.8 m/s^2)
Now, we can equate the equations for T and W_head:
1.07 * W_person = (mass of W) * (9.8 m/s^2)
We can now solve for the mass of W:
Mass of W = (1.07 * W_person) / (9.8 m/s^2)
Substituting the value of W_person (617.4 N), we can find the mass of W.