Question

consider the system pictured below. the cable can be considered massless, and the system is in static equilibrium. a) suppose the cable has maximum tension of 1200 n. if the beam is 50 kg and 3 m long, what is the value of d that corresponds to this tension? b) to prevent the cable from snapping, should d be higher or lower than the value in part a)? justify using your analysis from a).

Answer 1

(a) The value of
`\(d\)` that corresponds to the maximum tension of 1200 N in the cable is approximately 2.45 m.

(b) To prevent the cable from snapping,
`\(d\)` should be higher than the value in part (a). A higher
`\(d\)` would reduce the tension in the cable, ensuring it stays within the maximum tension limit of 1200 N.

Step-by-step explanation:

(a) In static equilibrium, the sum of torques acting on the beam is zero. The torque
`(\(\tau\))` can be calculated using the formula
`\(\tau = r \cdot F\),` where
`\(r\)` is the distance from the pivot point to the force application point, and
`\(F\)` is the force.

For part (a), when the cable has a maximum tension of 1200 N, the torque due to the tension in the cable
`(\(\tau_{\text{cable}}\))` balances the torque due to the gravitational force on the beam
`(\(\tau_{\text{gravity}}\))`. The equation for this equilibrium is
`\(r \cdot T = (1)/(2) \cdot m \cdot g \cdot d\),` where
`\(m\)` is the mass of the beam,
`\(g\)` is the acceleration due to gravity, and
`\(d\)` is the distance.

Using the known values (maximum tension
`\(T = 1200 \, \text{N}\),` mass of the beam
`\(m = 50 \, \text{kg}\),` length of the beam
`\(l = 3 \, \text{m}\),` and acceleration due to gravity
`\(g = 9.8 \, \text{m/s}^2\)),` we can rearrange the torque equation to solve for
`\(d\):`

`\[ d = (2 \cdot r \cdot T)/(m \cdot g) \]`

Substituting the values, we get:

`\[ d = \frac{2 \cdot 1.5 \, \text{m} \cdot 1200 \, \text{N}}{50 \, \text{kg} \cdot 9.8 \, \text{m/s}^2} \approx 2.45 \, \text{m} \]`

(b) Increasing
`\(d\)` provides a safety margin by reducing the tension in the cable. A higher
`\(d\)` ensures that even under variations or uncertainties, the tension does not exceed the maximum allowed (1200 N), enhancing the safety and integrity of the system.