Hospital patient. A 90.0 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?
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.
1. Weight (W): Since the patient is in a raised bed, there is a force acting downward due to gravity. This force is equal to the patient's mass (90.0 kg) multiplied by the acceleration due to gravity (9.8 m/s²). So, W = m × g, where g is approximately 9.8 m/s².
2. Tension force (T): The wire attached to the brace on the patient's neck exerts a force parallel to the bed, preventing the patient from sliding up. This tension force needs to be balanced by a force in the opposite direction to avoid sliding.
3. Friction force (f): The patient is subject to both kinetic and static friction between their body and the bed. The friction force opposes the motion or tendency to motion and acts parallel to the bed. The maximum friction force can be calculated using the equation f = μ × N, where μ is the coefficient of static friction (0.800 in this case) and N is the normal force.
In this scenario, the normal force (N) is equal to the weight (W) because the patient is not accelerating upwards or downwards. So, N = W = m × g.
To determine the maximum mass (m) without sliding, we need to find the maximum tension (T) that can be exerted without overcoming static friction. The maximum static frictional force is given by fs_max = μs × N.
Therefore, fs_max = μs × (m × g), where μs is the coefficient of static friction (0.800).
Since the tension force (T) is balanced by the maximum static frictional force (fs_max), we have T = fs_max = μs × (m × g).
Now we can equate T and the force due to friction (f) to determine the maximum mass (m) without sliding:
μk × (m × g) = μs × (m × g)
Dividing both sides by g, we get:
μk × m = μs × m
Simplifying further:
m(μk - μs) = 0
For the patient not to slide, the maximum mass (m) would be 0, which indicates that any mass less than this value would prevent sliding up the bed.
Now, let's calculate the patient's acceleration if the wire suddenly breaks. In this case, there would be no external force to counteract the patient's weight, so the net force acting on the patient would be the difference between the weight force (downward) and the kinetic friction force (upward).
The net force (F_net) = W - fk, where fk is the force due to kinetic friction.
Using the equation fk = μk × N, where μk is the coefficient of kinetic friction (0.500), the net force becomes:
F_net = W - μk × N
Substituting the values:
F_net = (90.0 kg × 9.8 m/s²) - (0.500 × (90.0 kg × 9.8 m/s²))
Finally, we can determine the patient's acceleration (a) using Newton's second law, which states that F_net = m × a:
a = F_net / m
Substituting the values, we can calculate the acceleration in m/s².
To find the maximum mass (m) that the patient can have without sliding up the bed, we need to consider the force of friction between the patient and the bed.
1. Determine the gravitational force acting on the patient:
F_gravity = mass * g, where mass = 90.0 kg and g = 9.8 m/s^2.
F_gravity = 90.0 kg * 9.8 m/s^2 = 882 N.
2. Determine the maximum static frictional force (F_static_max):
F_static_max = coefficient of static friction * normal force.
The normal force is equal to the gravitational force in this case because the patient is suspended.
F_static_max = 0.800 * F_gravity = 0.800 * 882 N = 705.6 N.
3. The maximum mass (m) that the patient can have without sliding up the bed is given by:
m = F_static_max / g.
Substituting the values:
m = 705.6 N / 9.8 m/s^2 = 72.0 kg (approximately).
Therefore, the maximum mass that the patient can have without sliding up the bed is approximately 72.0 kg.
Now, let's calculate the patient's acceleration if the wire suddenly breaks:
When the wire breaks, the only force acting on the patient will be the force of gravity (F_gravity). The patient's acceleration (a) can be determined using Newton's second law:
F_net = m * a,
where F_net is the net force acting on the patient and m is the patient's mass.
Since there are no other forces acting on the patient apart from gravity, the net force will be equal to the gravitational force:
F_net = F_gravity.
Therefore,
F_gravity = m * a.
Rearranging the equation:
a = F_gravity / m.
Substituting the values:
a = 882 N / 90.0 kg = 9.8 m/s^2.
Therefore, if the wire suddenly breaks, the patient's acceleration will be 9.8 m/s^2, which is equal to the acceleration due to gravity.