Final answer:
To calculate the cable tension Tb, one must account for the mass of the wrecking ball and the angle of the cable with the vertical. The tension is found by dividing the weight of the ball by the sine of the angle. The calculated tension ensures that the ball remains in static equilibrium.
Step-by-step explanation:
To calculate the tension Tb in the cable that makes an angle of 40° with the vertical when supporting a wrecking ball of mass m = 3520 kg, we must consider the forces acting on the ball while at rest (static equilibrium). The only forces acting are the gravitational force (weight of the wrecking ball) and the tension in the cable. The gravitational force is equal to the mass of the ball multiplied by the acceleration due to gravity (g = 9.81 m/s²), which is directed downwards. The tension in the cable has two components: a vertical component Tv that balances the weight of the wrecking ball and a horizontal component Th that will be in equilibrium with any horizontal forces (which are zero in this static case).
To find the vertical component Tv, we use the sine of the angle since it is opposite to the angle in the right triangle formed by Tb and its components. Thus, Tv = Tb * sin(φ), with φ being the given angle of 40°. The horizontal component Th would be Tb * cos(φ), but it is not relevant in this static scenario as there are no horizontal forces. Since the wrecking ball is in equilibrium, the vertical component of the tension must equal the weight of the wrecking ball, which means: Tv = m * g = Tb * sin(φ). To find the tension Tb, we rearrange for Tb = (m * g) / sin(φ). Substituting the given values, we get Tb = (3520 kg * 9.81 m/s²) / sin(40°).
After calculating this expression, we will obtain the value for the cable's tension Tb, which completes the solution for this physics problem. This type of problem is a classic example of static equilibrium and the decomposition of forces in physics