A 1.60 kg win bottle is lying horizontally in a rack. The two surfaces on which the bottle rests are 90.0° apart, and the right surface makes an angle of 45.0° with respect to the horizontal. Each surface exerts a force on the bottle that is perpendicular to the surface. Both forces have the same magnitude F. Find the value of F. Assume g = 9.81 m/s2.
To find the value of F, we can use the principles of equilibrium in rotational motion.
Let's break down the problem step-by-step:
1. Identify the forces acting on the wine bottle:
- The weight of the bottle acting vertically downwards (mg)
- The force exerted by the left surface (F_left)
- The force exerted by the right surface (F_right)
2. The weight of the wine bottle can be calculated using the formula:
weight (W) = mass (m) * acceleration due to gravity (g)
W = 1.60 kg * 9.81 m/s^2 = 15.696 N
3. Since the bottle is in equilibrium, the net force acting on it must be zero. Therefore, the vertical forces must cancel each other out:
F_left * cos(90°) + F_right * cos(45°) - W = 0
4. The horizontal forces must also balance each other out:
F_left * sin(90°) - F_right * sin(45°) = 0
Now we can solve these equations to find the value of F:
From the first equation:
F_left * cos(90°) + F_right * cos(45°) - W = 0
F_left * (0) + F_right * (cos(45°)) - 15.696 N = 0
F_right * (1 / √2) = 15.696 N
F_right = 15.696 N * √2
From the second equation:
F_left * sin(90°) - F_right * sin(45°) = 0
F_left * (1) - F_right * (sin(45°)) = 0
F_left - F_right * (1 / √2) = 0
F_left = F_right * (1 / √2)
Now substitute the value of F_right in terms of F_left from the second equation into the first equation:
F_right = 15.696 N * √2
F_left = F_right * (1 / √2)
F_left = (15.696 N * √2) * (1 / √2)
F_left = 15.696 N
Therefore, the value of F is approximately 15.696 N.
To find the value of F, we can resolve the gravitational force acting on the bottle into components parallel and perpendicular to the inclined surface.
1. Calculate the gravitational force acting on the bottle:
Fg = m * g
= (1.60 kg) * (9.81 m/s^2)
≈ 15.696 N
2. Resolve the gravitational force into components:
F_parallel = Fg * sin(45°)
= 15.696 N * sin(45°)
≈ 11.086 N
F_perpendicular = Fg * cos(45°)
= 15.696 N * cos(45°)
≈ 11.086 N
3. Since the two surfaces exert forces of the same magnitude F, F_parallel is equal to F.
Therefore, F = 11.086 N.
So, the magnitude of the force exerted by each surface on the bottle is approximately 11.086 N.