Problem 14 - Applied Physics

Question

The handle of a jackscrew is \(60.0 \mathrm{~cm}\) long. (a) If the mechanical advantage is \(78.0\), what is the pitch? (b) How much weight can be raised by applying a force of \(43 \overline{0} \mathrm{~N}\) to the jackscrew handle?

Step-by-Step Solution

Answer

(a) The pitch is approximately 4.84 cm. (b) 3354 N of weight can be raised.

Step 1: Understand the Relationship for Part (a)

The mechanical advantage (MA) for a screw jack is given by the formula: \(MA = \frac{2\pi \times \text{handle length}}{\text{pitch}}\). Here, the handle length is given as \(60.0\, \text{cm}\) and the mechanical advantage is \(78.0\). We need to find the pitch.

Step 2: Rearrange the Formula for Pitch

Rearrange the formula to solve for the pitch: \( \text{pitch} = \frac{2\pi \times \text{handle length}}{MA} \). Substitute the values to calculate:

Step 3: Substitute the Values to Calculate Pitch

Substitute the given values: \( \text{pitch} = \frac{2 \pi \times 60.0\, \text{cm}}{78.0} \). Calculate this to find the pitch.

Step 4: Calculate Pitch

Perform the calculation: \[ \text{pitch} = \frac{2 \times 3.1416 \times 60}{78} \approx 4.84\, \text{cm} \]. So, the pitch is approximately \(4.84\, \text{cm}\).

Step 5: Understand the Relationship for Part (b)

The force applied at the handle and the weight raised are related by the mechanical advantage. The weight \(W\) can be calculated using \(W = \text{Force} \times \text{MA}\). We have a force of \(43\, \text{N}\) and \(MA = 78.0\).

Step 6: Calculate the Weight Raised

Calculate the weight using the formula: \(W = 43\, \text{N} \times 78.0\). The weight raised will be \(43 \times 78 = 3354\, \text{N}\).

Key Concepts

Mechanical Advantage in Jackscrews

Mechanical advantage is a concept that describes how much a machine amplifies the input force to perform work. In the context of a jackscrew, the mechanical advantage (MA) helps us understand how much easier it is to lift a heavy object by applying a relatively smaller force. The mechanical advantage is calculated using the formula: - \[ MA = \frac{2\pi \times \text{handle length}}{\text{pitch}} \] Here, the handle length is the distance from the pivot where force is applied. In the given exercise, a jackscrew with a handle length of 60.0 cm has a mechanical advantage of 78. This means that the force applied at the handle is 78 times less than the force needed to lift the object without the jackscrew. It highlights the effectiveness of the machine in making tasks easier, illustrating why jackscrews are common in lifting heavy loads.
Mechanical advantage is not only a physical measure of the machine's power but also a crucial principle in mechanical engineering.

Pitch Calculation of a Jackscrew

To understand the mechanics of a jackscrew, it's crucial to calculate the pitch, which is the distance between the threads of the screw. The pitch measurement determines how far the screw will move vertically for every complete turn. To calculate the pitch using the given mechanical advantage, we rearrange the formula: - \[ \text{pitch} = \frac{2\pi \times \text{handle length}}{MA} \] With a handle length of 60.0 cm and a mechanical advantage of 78, substituting these values into the formula gives us: - \[ \text{pitch} = \frac{2 \cdot 3.1416 \cdot 60}{78} \approx 4.84\, \text{cm} \] Thus, the pitch is approximately 4.84 cm. This calculation shows how changing the pitch influences how much easier it is to lift the object with the jackscrew. A smaller pitch means the screw lifts less with each turn, but this corresponds to the higher mechanical advantage, making it easier to lift heavy loads.

Force and Weight Relationship in Jackscrews

Understanding the force and weight relationship is vital for solving physics problems involving jackscrews. The mechanical advantage allows us to relate the input force applied at the handle to the weight that can be lifted. This relationship is defined by the equation:- \[ W = \text{Force} \times MA \]Where \( W \) is the weight, \( \text{Force} \) is the force applied, and \( MA \) is the mechanical advantage. For a force of 43 N applied to a jackscrew handle with an MA of 78, the weight that can be lifted is: - \[ W = 43 \, \text{N} \times 78 = 3354 \, \text{N} \]So, the jackscrew can raise a weight of 3354 N. This illustrates how mechanical advantage allows for a smaller force to lift a significantly larger weight, making devices like jackscrews practical tools in mechanical applications.

Effective Physics Problem-solving with Jackscrews

Physics problem-solving often involves breaking down complex problems into simpler parts. When working with jackscrews, consider the relationship between force, distance, and mechanical advantage. Knowing how to rearrange formulas is a key skill. For example, to find the pitch, you'd start from the mechanical advantage formula and solve for the desired unknown. - Rearranging and substituting values helps make calculations straightforward. - Understanding the interplay of physical quantities aids in predicting machine behavior. - Visualizing the mechanical system can make concepts clearer. From these insights, solving problems with jackscrews becomes more manageable. These steps not only apply to jackscrews but also to a wide range of physics and engineering problems, emphasizing the importance of methodical approaches in problem-solving. By mastering these skills, students enhance their ability to tackle diverse mechanical problems effectively.