Question

The figure shows loads hanging from the ceiling of an elevator that is moving at constant velocity. Find the tension in each of the three strands of cord supporting each load, given that θ₁ =41°, θ₂ = 49°, θ₃ = 56°, m₁ = 5 kg, and m₂ = 8 kg.

Figure (a)

T1= ? T2 = ? T3 = ?

Figure (b)

T1 = ? T2= ? T3 = ?

Answer 1

Concepts and reason

The concepts required to solve the given question are Newton’s second law of motion, and the resolving forces.

First, apply Newton’s law of motion to find the horizontal and vertical forces acting in this case. Arrive at an expression in terms of the tension acting on the cords. Then rearrange and simplify the expressions to calculate the tension acting on the first and second cords.

To find the tension acting on the third cord, calculate the weight of the load. In the next part, apply Newton’s law to find the horizontal and vertical components of force acting. Solve the equations to find the tension acting in cord 1 and 2. Tension in the third cord can be calculated from the weight of the second load.

Fundamentals

Tension acting in the cord supporting the load:

Force acting on a body may be defined as the sudden push or pull which is capable of changing the direction of motion of the object if it is unopposed.

Newton’s second law states that the acceleration of an object produced by a net force is directly proportional to the magnitude of net force and inversely proportional to the mass of the object.

Mathematical expression for Newton’s law is given by,

Here, F is the force acting; m is the mass and is the acceleration.

Weight of an object is the “force of gravity”. It is defined as the product between the mass and the acceleration due to gravity.

Weight of the object is,

Here, W is the weight of the object, m is the mass of the object and g is the acceleration due to gravity.

Tension force is defined as the “force that is transmitted through a string, rope cable or wire when it is pulled tight by forces acting from opposite ends”. It is directed along the length of the wire and it pulls the objects equally on opposite ends of the wire.

(a.1)

Tension acting in the cords supporting the load having mass of 5 kg:

Apply Newton’s second law of motion; the horizontal components of force acting are,

Substitute for and for.

…… (1)

Vertical components of force acting,

Rewrite the expression in terms of gravity.

Substitute 5 kg for , for g, for , for.

…… (2)

Substitute for .

(a.2)

Tension acting in cord 2 is,

Substitute 32.15 N for .

(a.3)

Tension acting in the third cord,

Substitute 5 kg for and for g.

(b.1)

Tension acting in the cords supports the load having a mass of 8 kg:

Applying Newton’s second law of motion, horizontal components of force acting is,

Substitute and rewrite the equation.

Substitute for .

…… (3)

Vertical component of force acting is,

Substitute and rewrite the equation.

…… (4)

Substitute 8 kg for and for g and for .

(b.2)

Tension in the cord 2 is given by,

Substitute 94.56 N for .

(b.3)

Tension acting in the cord 3 is,

Rewrite the expression for weight.

…… (5)

Substitute 8 kg for and for g.

Ans: Part a.1

Tension acting in the cords supporting load 1 is.

Part a.2

The tension acting in the cords supporting load 1 is .

Part a.3

Tension acting in the cords supporting load 1 is .

Part b.1

Tension acting in the cords supporting load 2 is .

Part b.2

Tension acting in the cords supporting load 2 is .

Part b.3

Tension acting in the cords supporting load 2 is .