91Ó°ÊÓ

Calculating force is an essential part of understanding how objects interact in physics. Force is generally calculated using Newton's second law of motion, which is given by the formula \[F = m \cdot g\]where:

• \(F\) is the force in newtons (N)
• \(m\) is the mass of the object in kilograms (kg)
• \(g\) is the acceleration due to gravity, approximately \(9.8\, \text{m/s}^2\) on Earth

In the context of the climbing rope exercise, the force exerted by the climber on the rope is calculated by multiplying the climber's mass by the acceleration due to gravity. Therefore, the force \[F = 90 \text{ kg} \times 9.8 \text{ m/s}^2 = 882 \text{ N}\].This tells us that the rope will have to withstand a force of 882 newtons, providing vital information for assessing its strength and elasticity.

Understanding the area of a circle is crucial when dealing with objects like ropes or wires with circular cross-sections. The area is calculated using the formula:\[A = \pi r^2\]where:

• \(A\) is the area
• \(\pi\) is Pi, approximately equal to 3.14159
• \(r\) is the radius of the circle

For the climbing rope problem, the diameter of the rope is given as 1.0 cm, so the radius is half of the diameter: \[r = \frac{1.0 \text{ cm}}{2} = 0.5 \text{ cm} = 0.005 \text{ m}\].Substituting the radius into the area formula gives us:\[A = \pi (0.005 \text{ m})^2 \approx 7.85 \cdot 10^{-5} \text{ m}^2\].This area represents the cross-section through which the force is distributed, and it's a key factor in calculating Young's modulus, or the elasticity of the rope.

Elasticity refers to the ability of a material to resume its normal shape after being stretched or compressed. In physics, Young's modulus quantifies this property. To compute Young's modulus, we use the formula:\[Y = \frac{F/A}{\Delta L/L}\]where:

• \(Y\) is Young's modulus
• \(F\) is the force applied
• \(A\) is the cross-sectional area
• \(\Delta L\) is the change in length (elongation) of the material
• \(L\) is the original length of the material

For our rope, the force applied is 882 N, and the area is \(7.85 \times 10^{-5} \text{ m}^2\). The elongation \(\Delta L\) is given as 1.6 m, and the original length \(L\) is 50 m. Substituting these values into the formula gives:\[Y = \frac{882 \text{ N} / 7.85 \times 10^{-5} \text{ m}^2}{1.6 \text{ m} / 50 \text{ m}} = 1.40 \times 10^{11} \text{ N/m}^2\].Young's modulus allows us to assess the rope's elasticity or stiffness, indicating how much it will stretch under a specified load.