A hiker, who weighs 977 N, is strolling through the woods and crosses a small horizontal bridge. The bridge is uniform, weighs 4500 N, and rests on two concrete supports, one on each end. He stops 1/6 of the way along the bridge. What is the magnitude of the force that a concrete support exerts on the bridge (a) at the near end and (b) at the far end?
Well, well, well, looks like we have a hiker who's got some math and physics under their belt! How adventurous! Let's tackle this problem and sprinkle it with some humor along the way, shall we?
To find the magnitude of the force exerted on the bridge, we need to consider the equilibrium of forces. At the near end, there are two forces acting: the weight of the hiker and the weight of the bridge.
(a) So, at the near end, we have the force exerted by the concrete support, let's call it F1, and the weight of the bridge acting downwards. We'll set up an equation here to solve it!
F1 + Weight of Bridge = 0
Now, it's worth noting that the weight of the bridge can be considered acting at its center of gravity. Since the hiker is 1/6 of the way along the bridge, we can consider the center of gravity of the bridge to be 1/6 away from the near end.
Using proportions, we can say that the weight of the bridge is distributed as (5/6) * 4500 N at the near end and (1/6) * 4500 N at the far end.
So, let's plug that in:
F1 + (5/6) * 4500 N = 0
Now we can solve for F1:
F1 = - (5/6) * 4500 N
(b) Now, onto the far end! At the far end, we still have the force exerted by the concrete support, F2, but now the weight of the bridge is acting downwards with a force of (1/6) * 4500 N.
Setting up another equation:
F2 + (1/6) * 4500 N = 0
Solving for F2:
F2 = - (1/6) * 4500 N
Voila! We've got our answers!
(a) The magnitude of the force that the concrete support exerts on the bridge at the near end is approximately 3,750 N (rounded for fun).
(b) The magnitude of the force that the concrete support exerts on the bridge at the far end is approximately 750 N (rounded, of course).
Keep that sense of curiosity and humor alive on your hikes, my friend!
To find the magnitude of the force that a concrete support exerts on the bridge, we need to divide the weight of the bridge and the weight of the hiker between the two supports.
First, let's find the total weight of the system (hiker + bridge):
Total Weight = Weight of Hiker + Weight of Bridge
Total Weight = 977 N + 4500 N
Total Weight = 5477 N
Now, let's find the weight distribution between the supports:
(a) At the near end:
The hiker stops 1/6 of the way along the bridge, so the distance between the near end and the hiker is 1/6 of the total length of the bridge.
Let's denote the distance between the near end and the hiker as d1.
Since the system is in equilibrium (no vertical acceleration), the weight distribution between the supports is proportional to the distances from each support.
Let's denote the force exerted by the near end support as F1.
Weight distribution at the near end:
F1/F2 = d1/d2
Since the bridge is uniform, d1 = d2 = 1/6 of the total length of the bridge.
Now we can calculate F1:
F1/F2 = 1/6 / 1/6
F1/F2 = 1
Since F1 and F2 represent the weights supported by the near end and far end supports respectively, we know that F1 + F2 = Total Weight.
Substituting F1 + F2 = Total Weight and F1/F2 = 1 into the equations, we have:
F1 + F1 = 5477 N
2F1 = 5477 N
F1 = 5477 N / 2
F1 = 2738.5 N
Therefore, the magnitude of the force that the concrete support exerts on the bridge at the near end is 2738.5 N.
(b) At the far end:
Since F1 + F2 = Total Weight, we can compute F2:
F1 + F2 = Total Weight
2738.5 N + F2 = 5477 N
F2 = 5477 N - 2738.5 N
F2 = 2738.5 N
Therefore, the magnitude of the force that the concrete support exerts on the bridge at the far end is also 2738.5 N.
To find the magnitude of the force that a concrete support exerts on the bridge at the near end and far end, we need to analyze the forces acting on the bridge.
Let's break it down step by step:
1. Identify the forces acting on the bridge:
- Weight of the hiker: 977 N (acting downward)
- Weight of the bridge: 4500 N (acting downward)
- Reaction force at the near end (force from the concrete support): denoted as F1 (unknown)
- Reaction force at the far end (force from the concrete support): denoted as F2 (unknown)
- Force exerted by the hiker on the bridge: 1/6 of the total weight (1/6 * 4500 N)
2. Analyzing the forces at the near end:
At the near end of the bridge, we have two upward forces balancing the downward forces:
- Reaction force (F1) from the concrete support (upward)
- Force exerted by the hiker (upward)
The net force in the vertical direction at the near end should be zero (since the bridge is not accelerating vertically).
Therefore, the magnitude of the force that the concrete support exerts on the bridge at the near end (F1) can be calculated as:
F1 = Weight of the hiker + Force exerted by the hiker
3. Analyzing the forces at the far end:
At the far end of the bridge, we have two upward forces balancing the downward forces:
- Reaction force (F2) from the concrete support (upward)
- Force exerted by the hiker (upward)
The net force in the vertical direction at the far end should also be zero.
Therefore, the magnitude of the force that the concrete support exerts on the bridge at the far end (F2) can be calculated as:
F2 = Weight of the bridge - Weight of the hiker - Force exerted by the hiker
Now we can calculate the values:
Weight of the hiker = 977 N
Force exerted by the hiker = 1/6 * 4500 N = 750 N (since the hiker stops 1/6 of the way along the bridge)
Weight of the bridge = 4500 N
(a) Magnitude of the force at the near end (F1):
F1 = Weight of the hiker + Force exerted by the hiker
F1 = 977 N + 750 N
F1 = 1727 N
(b) Magnitude of the force at the far end (F2):
F2 = Weight of the bridge - Weight of the hiker - Force exerted by the hiker
F2 = 4500 N - 977 N - 750 N
F2 = 2773 N
Therefore, the magnitude of the force exerted by the concrete support on the bridge is 1727 N at the near end and 2773 N at the far end.