Question

A hook above a stage is rated to support $150.$ lb. A 3 -lb rope is attached to the hook, and a 147 -lb actor is going to attempt to swing across the stage on the rope. Will the hook hold the actor up during the swing?

Short answer

Answer: Yes, the hook can support the combined weight of the rope and the actor during the swing.

Step-by-step solution

Determine the total weight of the rope and actor

First, we need to find the total weight of the rope and actor. To do this, we simply add the weight of the rope (3lb) to the weight of the actor (147lb). So, the total weight is: Total weight = Weight of rope + Weight of actor

Calculate the total weight

Now we can insert the given values and calculate the total weight. Total weight = 3lb + 147lb Total weight = 150lb

Compare the total weight to the hook's weight limit

Now, we need to compare the total weight (150lb) to the weight limit of the hook (also 150lb). We want to know if the hook can support this weight, so we must see if the total weight is less than or equal to the weight limit of the hook.

Determine if the hook can hold the actor up during the swing

Since the total weight (150lb) is equal to the weight limit of the hook (150lb), the hook will be able to hold the actor up during the swing. The hook will be under its maximum rated weight, and thus should be able to safely support the actor and rope.

Key concepts

Weight Calculation

Understanding weight calculation is crucial when determining the stress and maximum load capacity of objects in everyday scenarios. Calculating weight involves summing up individual weights of all components involved. For instance, in this problem, we need to find the weight exerted by both the rope and the actor acting on the hook.

• The rope weighs 3 pounds.
• The actor weighs 147 pounds.

Adding these together, we arrive at a total weight: Total weight = 3 lb (rope) + 147 lb (actor) = 150 lb. It’s essential to perform these calculations prior to placing any load on a supporting structure to ensure safety and prevent overloading.

Force and Weight Limits

Force is the push or pull acting upon an object. When considering weight limits, it's important to determine whether the force exerted by a mass (in this case, the combined weight of the rope and the actor) exceeds the maximum capacity of the object supporting it—here, the hook. Generally, this maximum capacity is referred to as the weight limit. To assess the hook's ability:

• We calculate the total weight: 150 lb.
• We compare this with the hook's weight limit: 150 lb.

If the load is equal to or less than the weight limit, the structure is considered safe. In our case, the hook can hold 150 lb, which matches the combined weight of the rope and actor. Although this indicates the hook should hold, it’s important to consider safety margins, especially if dynamic or additional forces come into play during the swing, such as increased tension.

Static Equilibrium

Static equilibrium occurs when all forces acting on a system balance out such that the system remains stable and at rest. In terms of a hanging object like our stage scenario, this means the upward forces need to equal the downward forces. For the hook to maintain static equilibrium:

• Downward force (weight of rope and actor) must equal the upward force (capacity of the hook).
• If both are balanced, as they are here at 150 lb each, the actor can safely swing.

However, it's essential to note that equilibrium doesn't guarantee safety if close to limits. External factors such as sudden jerks or swings could introduce additional forces, taking the setup out of equilibrium. Therefore, engineers often design systems with safety factors beyond static equilibrium conditions to assure reliability under various conditions.