Question

The century-old ascensores in Valparaiso, Chile, are small cable cars that go up and down the steep hillsides. As the figure shows, one car ascends as the other descends. The cars use a two-cable arrangement to compensate for friction; one cable passing around a large pulley connects the cars, the second is pulled by a small motor. Suppose the mass of both cars (with passengers) is 1200, the coefficient of rolling friction is 3.0×10−2, and the cars move at a constant speed.

a. What is the tension in the connecting cable?
b. What is the tension in the cable to the motor?

Answer 1

Let us model the two cable cars to be particles, the cable’s to be massless and the pulleys to be massless and frictionless.

We need to find the tension in the two cables moving the two cable cars up and down. Since the two cable cars are moving with constant velocity, the net force on each cable car should be zero.

Thus, our task will be to identify all the forces acting on the cable cars, resolve them in two perpendicular directions and equate the net force in each direction to zero.

The figure below shows the forces acting on each cable car.

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T is the tension in the cable that joins the two cables cars together. T_m is the tension in the cable connecting the cable car to the motor. mg is the weight of each cable car.

The frictional force acts in the opposite direction of motion of each cable car. F_u is the frictional force on the cable car which is going up while F_d is the friction on the cable car which is going down.

R _u, R_d are the normal forces on the cable cars moving up and down respectively.

Let’s resolve forces on the cable car which is going down in two perpendicular directions, downward along the slope, upward perpendicular to the slope and equate each to zero.

Perpendicular to the slope:

…… (1)

Down the slope:

…… (2)

Resolving forces on the cable car going up in two perpendicular directions and equating each to zero:

Perpendicular to the slope:

…… (3)

Down the slope:

…… (4)

(a)

From equation (1),

The frictional force is,

Here, is the rolling friction coefficient.

Substitute for .

From the above two equations, we get

…… (5)

Plugging for in equation (2) from equation (5) gives,

Rearranging the above equation for.

Substituting for m, for g, and for.

Hence, the tension in the connecting table is.

(b)

From equation (3) we can obtain

…… (6)

Once again, we can relate to the normal reaction from the slope through

…… (7)

Substitute for .

…… (8)

Plugging for in equation (4).

Solving for T_m.

Substituting for m, for g, for, and for T.

Therefore, the tension in the cable to the motor is .