We consider a man of mass m = 77 kg as shown in the figure below using crutches. The crutches each make an angle of è = 26 with the vertical. Half of the person's weight is supported by the crutches, the other half is supported by the normal forces acting on the soles of the feet. Assuming that the person is at rest, find the magnitude of the force supported by each crutch.
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m=77kg
angle=26 degrees
Normal force holds half the weight.
Positive being up,
Y direction:
(1/2)w-(F1)cos(26)-(F2)cos(26)
=(1/2)(77kg)(9.8m/s^2)-2Tcos(26) [since T1=T2
=> T=377.3N/2cos(26)
T=210N
m=77kg
angle=26 degrees
Normal force holds half the weight.
Positive being up,
Y direction:
(1/2)w-(F1)cos(26)-(F2)cos(26)
=(1/2)(77kg)(9.8m/s^2)-2Fcos(26) [since F1=F2
=> F=377.3N/2cos(26)
F=210N
To find the magnitude of the force supported by each crutch, we can start by analyzing the forces acting on the person.
Let's break down the forces acting on the person:
1. Weight (mg): The force due to gravity acting vertically downward. Its magnitude is equal to the mass of the person (m) multiplied by the acceleration due to gravity (g ≈ 9.8 m/s²).
Weight (mg) = m * g
2. Normal forces (N₁, N₂): The normal forces acting on the soles of the feet in the upward direction. They support half of the person's weight each, so the magnitude of each normal force is equal to half of the person's weight.
Normal force (N₁) = Normal force (N₂) = (1/2) * (mg)
3. Forces supported by crutches (F₁, F₂): The forces supported by each crutch and acting upwards along the inclined direction. We need to find their magnitude.
Now, let's apply trigonometry to find the magnitude of the forces supported by each crutch.
Since each crutch makes an angle of è = 26° with the vertical, the horizontal component of the respective force (F₁, F₂) will be given by:
Horizontal component = Normal force * sin(è)
Similarly, the vertical component of the respective force (F₁, F₂) will be given by:
Vertical component = Normal force * cos(è)
Since the person is at rest, the vertical components of the two forces (F₁, F₂) must balance the weight (mg), and the horizontal components of the two forces must cancel each other out.
Therefore, we can equate the vertical components and horizontal components to find the magnitude of the forces supported by each crutch:
Vertical components:
F₁ * cos(è) + F₂ * cos(è) = (1/2) * (mg)
Horizontal components:
F₁ * sin(è) - F₂ * sin(è) = 0
Now, we can solve these two equations simultaneously to find the magnitudes of the forces supported by each crutch.