Question

A hook above a stage is rated to support 150 . Ib. A 3.00-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 actor during the swing, as the total weight on the hook (150 lbs) is equal to its rated capacity.

Step-by-step solution

Calculate the total weight on the hook

To calculate the total weight on the hook, sum up the weight of the rope and the actor. The rope weighs 3.00 lb and the actor weighs 147 lb. Total weight = weight of rope + weight of actor = 3.00 lb + 147 lb = 150 lb

Compare the total weight with the hook's rating

Now that we have the total weight acting on the hook, we need to compare it to the hook's rating. The hook is rated to support 150 lb. As the total weight (150 lb) is equal to the hook's rating (150 lb), the hook will be able to support the actor during the swing.

Key concepts

Weight Calculation

The concept of weight calculation is fundamental in physics, especially when discussing the stability of structures or the safety of objects in suspension. Weight can be thought of as the force exerted by gravity on an object and is proportional to the object's mass. To calculate it, we typically use the formula:
\( W = m \times g \), where \( W \) is the weight, \( m \) is the mass, and \( g \) is the acceleration due to gravity, which is approximately 9.8 m/s² on Earth. However, in many common problems, especially in the United States, weight is often given in pounds (lb), which is a unit of force.

Importance of Accurate Weight Assessment

One critical factor to keep in mind is the accuracy of weight measurements. In scenarios like the provided exercise, knowing the precise weight of both the rope and the actor is essential for ensuring safety. If the calculated total weight is at or below the capacity of the hook, as in our case, it indicates that the hook can support the combined load. Notably, it's always advised to have a safety margin, as static weight differs from dynamic forces that come into play during motion.

Tension in Physics

Tension is a force that is transmitted through a string, cable, or rope when it is pulled tight by forces acting from opposite ends. It is an axial force that passes through the length of the medium and is directed along the wire or rope, away from the object applying the force. Tension is usually measured in Newtons (N) but can also be measured in pounds (lb) in certain regions.

Understanding Tension in Objects

When an actor swings across the stage on a rope, the rope will experience tension. This tension must be strong enough to hold the force due to the weight of the actor and any additional forces from the motion. The maximum tension that the rope can handle without breaking depends on its material and thickness. In the example problem, since the sum of the weights does not exceed the hook's rating, we can infer that the rope's tension is within its capacity if the rope's strength is not less than the hook's rating.

Forces and Motion

The interplay of forces and motion is at the heart of many physics problems. Forces are pushes or pulls that can cause an object to accelerate, decelerate, remain in place, or change direction. Newton's laws of motion are the foundation for understanding how forces affect the motion of objects.For simplicity, we often begin with scenarios assuming no other forces are acting on the system except gravity and tension, paving the pathway to more complex situations involving friction, air resistance, and other dynamic forces.

Dynamic Forces in Motion

In the context of our exercise, as the actor swings across the stage on a rope, the force of tension in the rope and the force of gravity are not the only forces at play. When the actor is moving, there are also dynamic forces due to the actor's acceleration and possible changes in velocity. If the motion involves complex paths, like circles or pendulums, centripetal forces must be considered. It is essential to ensure that the hook's rating and the rope's tension can handle these dynamic forces, not just the static weight of the actor. This concept highlights the importance of considering all aspects of motion when solving physics problems related to forces.