A tow truck is connected to a 1500 kg car by a cable that makes a 27 degree angle to the horizontal. If the truck accelerates at 0.58 m/s^2, what is the magnitude of the cable tension? Neglect friction and the mass of the cable.

Wc = mg = 1500kg * 9.8N/kg = 14,700 N. = Wt. of car.

Fc = 14700N @ 0 Deg. = Force of car.
Fp=14,700*sin(0) = 0 = Force parallel to road.
Fv = 14,700*cos(0) = 14,700 N. = Force
perpendicular to road.

Fap*cos27 = ma = 1500*0.58 = 870 N.
Fap = 870 / cos27 = 976.4 N.=Force applied = Tension in rope.

Use only the INFO in the last 2 lines.

The other INFO is not neded.

To find the magnitude of the cable tension, we need to consider the forces acting on the car.

1. The weight of the car acts vertically downward, and its magnitude can be calculated using the formula:
Weight = mass * gravitational acceleration
The gravitational acceleration is approximately 9.8 m/s^2.

Weight of the car = 1500 kg * 9.8 m/s^2

2. The tension in the cable can be broken into two components: one parallel to the incline and another perpendicular to the incline.

The component parallel to the incline causes acceleration, and the component perpendicular to the incline counteracts a part of the weight of the car.

Given:
Angle of the cable with the horizontal = 27 degrees
Acceleration of the truck = 0.58 m/s^2

To find the component of tension parallel to the incline:
Parallel component = Tension * sin(angle)

To find the component of tension perpendicular to the incline:
Perpendicular component = Tension * cos(angle)

Now, let's calculate the two components of tension:

Parallel component = T * sin(27 degrees)

Perpendicular component = T * cos(27 degrees)

According to Newton's second law, the net force acting on the car in the parallel direction is equal to the mass times the acceleration:

Net force = mass * acceleration

The net force acting on the car in the parallel direction is the difference between the parallel component of tension and the force due to gravity:

Net force = Parallel component - Weight of the car

Substituting the values, we have:

Parallel component - (1500 kg * 9.8 m/s^2) = (1500 kg * 0.58 m/s^2)

Simplifying the equation, we get:

T * sin(27 degrees) - (1500 kg * 9.8 m/s^2) = (1500 kg * 0.58 m/s^2)

Now, we can solve this equation to find the magnitude of the cable tension (T).

To find the magnitude of the cable tension, we need to consider the forces acting on the car. In this case, there are two main forces: the force of gravity acting downwards and the tension force in the cable acting upwards.

First, let's calculate the force of gravity on the car using the formula:

Force of gravity = mass × acceleration due to gravity

The mass of the car is given as 1500 kg, and the acceleration due to gravity is approximately 9.8 m/s^2. Therefore:

Force of gravity = 1500 kg × 9.8 m/s^2 = 14,700 N

Since the car is connected to the tow truck by a cable, there must be an opposing force in the cable that balances out the force of gravity acting downwards. This opposing force is the tension force in the cable.

To calculate the tension force, we need to consider the vertical component of the cable tension. The vertical component of the tension force can be calculated using the formula:

Tension force vertical = Force of gravity × sin(angle of cable)

In this case, the angle of the cable is given as 27 degrees. Therefore:

Tension force vertical = 14,700 N × sin(27 degrees) ≈ 6558.04 N

Now that we have the vertical component of the tension force, we can calculate the magnitude of the tension force using the formula:

Magnitude of the tension force = Tension force vertical / acceleration of the truck

The acceleration of the truck is given as 0.58 m/s^2. Therefore:

Magnitude of the tension force = 6558.04 N / 0.58 m/s^2 ≈ 11,300 N

So, the magnitude of the cable tension is approximately 11,300 N.