Equilibrium in 2D Practice Problems

Find the tension in cords A, B, and C for an object with weight w suspended by cords as shown below. 

A container being lowered from a crane is held in place by a light steel cable that makes an angle of 58.0° from the vertical and a horizontal synthetic cable. If the container has a mass m = 4520 kg, determine
a) The tension T₁ in the slanted cable.
b) The tension T₂ in the horizontal cable.

An 850-kg car is to be loaded on a car transporter. A 22.0° ramp measured above the horizontal is used for the process. A light cable making an angle of 28.0° above the surface of the ramp is used to hold the car temporarily stationary on the ramp. Find the tension in the cable if the ramp is very smooth (frictionless).

A crane lifts a slab using two steel cables attached to its upper edges. The tension in each cable is equal to 0.80 of the slab's weight. If the two cables have the same angle relative to the vertical, determine the value of this angle that makes the tension in the cable equal to 0.8 of the slab's weight.

The practice of slowly and gently pushing on a fractured or dislocated body part is known as traction. Often, ropes, pulleys, and weights are used for it. Consider a traction setup as shown in the figure. Assuming the pulleys are frictionless calculate the tension force in the rope that supports the leg.

Two traffic lights are installed at an intersection. One is put in place for traffic and weighs 12 kg, while the other is installed for pedestrians and weighs 8.0 kg. The figure shows the two traffic lights hanging and connected using a highly flexible and lightweight cable. Determine the tension and angle (θ₁) of the first cable if the center cable is adjusted to be perfectly horizontal.

Two physics enthusiasts devise an experiment in which a 40-centimeter-long thread suspends a bar magnet(m = 10g). When another bar magnet is brought close to the first one with the identical poles facing each other (north to north or south to south),  they repel one another. If the second magnet is held in place it causes the hanging magnet to swing out to a 15-degree angle and remain there. Determine the magnitude of the magnetic force that causes the repulsion between magnets (making it swing).

At a carnival, a challenge is set up for two people. To win the challenge they must pull a rectangular box to the finish line but both of them should have equal pulls. The angle between the two ropes tied to the box is 30 degrees. If the friction force on the box is 970 N, determine how much force each person must use to pull the box at a constant 3.0 m/s.

A construction worker needs to lift a heavy beam weighing 500 kg from a pile of beams on the ground to a height of 8.0 m using a pulley system. The plan is to attach a rope to the beam and then pull horizontally on the rope as shown in the figure below. When the worker pulls horizontally on the rope, the beam rises vertically. How much force must the worker exert horizontally on the rope to lift the beam 2.0 m above its initial position?

A weightlifter is engaging in a workout routine, aiming to lift a 95-kg iron block with the aid of a rope, as depicted in the figure. Determine the necessary downward force, represented as $\overrightarrow{F}$, that the weightlifter must exert to ensure the rope sags at its midpoint by 1.8 m.

A box is being pulled by three ropes as illustrated below. Given the known forces $\overrightarrow{F_1}$ = 200 N and $\overrightarrow{F_2}$ = 250 N, calculate the magnitude and direction of $\overrightarrow{F_3}$ necessary to maintain equilibrium.

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Two lamp posts are positioned 10.5 m apart on a street. A decorative sign of mass 25 kg is to be hung between these posts on a wire, with the goal of having the wire sag by 0.25 m at its midpoint to create an aesthetic droop. Determine the magnitude of the force that must be exerted downward at the midpoint of the wire to keep the sign in the desired position.

Calculate the radius of a straight steel wire that bends to form a 15° angle with the horizontal when a 35 kg mass is suspended from its midpoint. For steel, its Young's modulus $E=200 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}$.