The picture shows a uniform ramp between two buildings that allows for
motion between the buildings due to strong winds. At its left end it is
hinged to the building wall , at its right end it has a roller that can
roll along the building wall. There is no vertical force on the roller
from the building, only a horizontal force with magnitude Fh . The
horizontal distance between the buildings is D=4.00 m . The rise of the
ramp is h= 0.490m. A man walks across the ramp from the left Picture 5-1
gives Fh as a function of horizontal distance x of the man from the
building at the left. What are the masses of (a) the ramp and (b) the
man ?
To determine the masses of the ramp and the man, we need to analyze the forces acting on each of them individually.
Let's start with the ramp:
1. The only force acting horizontally on the ramp is the horizontal force, Fh, provided by the man. This force creates a torque about the hinge point, which allows the ramp to rotate.
2. The torque equation is given by: Torque = Force * Distance * sin(θ).
3. In this case, the force is Fh, the distance is the horizontal distance, D, between the buildings, and the angle, θ, is the angle of inclination of the ramp.
4. The angle of inclination, θ, can be calculated using the rise, h, and the horizontal distance, D, using the formula: θ = atan(h/D).
5. Therefore, the torque equation becomes: Torque = Fh * D * sin(atan(h/D)).
6. The torque provided by the man should be equal to the torque required to balance the ramp. This torque is given by: Torque = Mass * gravitational acceleration * Distance of the center of mass from the hinge * sin(θ).
7. Since the ramp is uniform, the center of mass is at its midpoint, which is h/2 above the hinge.
8. Combining equations, we get: Fh * D * sin(atan(h/D)) = Mass of ramp * g * (h/2) * sin(atan(h/D)).
Now, let's move on to the man:
1. The man applies a horizontal force, Fh, to the ramp, allowing it to balance against the wind.
2. The force exerted by the man on the ramp is equal in magnitude but opposite in direction to the horizontal force from the ramp on the man.
3. According to Newton's second law, Force = mass * acceleration.
4. Since the man is walking across a horizontal surface, his acceleration is zero, and thus, the force on the man is also zero.
5. Therefore, the mass of the man is zero.
In summary, the mass of the ramp can be calculated by setting the torque provided by the man equal to the torque required to balance the ramp. The mass of the man is zero since there is no net force acting on him.