A 71.0 kg patient is suspended in a raised hospital bed as shown in the figure. The wire is attached to a brace on the patient's neck and pulls parallel to the bed, and the coefficients of kinetic and static friction between the patient and the bed are 0.500 and 0.800, respectively.
What is the maximum mass m can be without the patient sliding up the bed?
If the wire suddenly breaks, what's the patient's acceleration?
The angew shown in the figure is 50 degrees from the hospital bed and the ground.
it is important where the wire is attached, and how the angle is measured.
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To solve this problem, we need to analyze the forces acting on the patient in both situations.
First, let's consider the situation where the patient does not slide up the bed. This means the force of kinetic friction between the patient and the bed is equal to the force produced by the wire pulling parallel to the bed.
1. Determine the force of kinetic friction (fk):
The force of kinetic friction is given by the equation fk = μk * N, where μk is the coefficient of kinetic friction and N is the normal force.
The normal force is equal to the gravitational force acting on the patient, which is equal to the patient's weight. So, N = m * g, where m is the mass of the patient and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Therefore, fk = μk * m * g.
2. Determine the force produced by the wire (Fwire):
The force produced by the wire is along the direction parallel to the bed. We can resolve this force into two components: one perpendicular to the bed (Fwire_perpendicular) and one parallel to the bed (Fwire_parallel).
The angle between the wire and the bed (θ) is given as 50 degrees. So, Fwire_parallel = Fwire * sin(θ).
Since the patient is in equilibrium, Fwire_parallel is equal to the force of kinetic friction (fk). Therefore, Fwire_parallel = fk.
3. Determine the mass m:
We know that Fwire_parallel = fk = μk * m * g.
Rearranging this equation to solve for m, we have:
m = Fwire_parallel / (μk * g).
Now let's consider the situation where the wire suddenly breaks. In this case, the only force acting on the patient is the force of gravity.
4. Determine the patient's acceleration (a) when the wire breaks:
The force of gravity acting on the patient is m * g.
Using Newton's second law (F = m * a), we can equate the force of gravity to the patient's mass times the acceleration, resulting in:
m * g = m * a.
Therefore, the patient's acceleration (a) when the wire breaks is equal to the acceleration due to gravity (g).
Using the given values and equations, we can now calculate the maximum mass (m) and the patient's acceleration (a) when the wire breaks.
To find the maximum mass m that the patient can have without sliding up the bed, we need to consider the forces acting on the patient in the vertical and horizontal directions.
In the vertical direction, we have the weight of the patient (mg) acting downward, and the normal force (N) exerted by the bed acting upward. Since the patient is not moving vertically, these forces must balance each other:
mg = N
In the horizontal direction, we have the force of friction (f) exerted by the bed, which opposes the force of the wire (T). The force of friction can be calculated using the coefficient of static friction (µs) and the normal force (N):
f = µs * N
The force of the wire can be resolved into its horizontal and vertical components. The horizontal component of T is T * cosθ, where θ is the angle shown in the figure (50 degrees). The weight of the patient can also be broken down into its horizontal component, which is mg * cosθ.
Now, we can set up equilibrium equations in the horizontal direction to find the maximum mass m:
T * cosθ = f
T * cosθ = µs * N
T * cosθ = µs * mg
Substituting the value of N = mg, we get:
T * cosθ = µs * mg
Solving for the mass m:
m = T * cosθ / (µs * g)
Now, to find the patient's acceleration if the wire suddenly breaks, we need to consider the forces acting on the patient in the vertical direction. The only force acting on the patient in this direction is the weight (mg) acting downward. Since there is no upward force to balance it, the patient will accelerate downward.
Therefore, the patient's acceleration would be equal to the acceleration due to gravity, which is approximately 9.8 m/s^2.