As shown in Fig. 3-11(a), an object of mass m is supported...

As shown in Fig. $3-11(a)$, an object of mass $m$ is supported by a cord. Find the tension in the cord if the object is $(a)$ at rest, $(b)$ moving at constant velocity, ( $c$ ) accelerating upward with acceleration $a=3 g / 2$, and $(d)$ accelerating downward at $a=0.75 g$. Two forces act on the object: the tension $F_{T}$ upward and the downward pull of gravity $m g$. They are shown in the free-body diagram in Fig. $3-11(b) .$ As a rule begin your analysis with a diagram. Take up as the positive direction and write $+\Sigma F_{y}=m a$, in each case. Notice that the tension in the cord is less than $m g$ in part $(d)$; only then can the object have a downward acceleration. Can you explain why $F_{T}=0$ if $a_{y}=-g$ ?

As shown in Fig. $3-11(a)$, an object of mass $m$ is supported by a cord. Find the tension in the cord if the object is $(a)$ at rest, $(b)$ moving at constant velocity, ( $c$ ) accelerating upward with acceleration $a=3 g / 2$, and $(d)$ accelerating downward at $a=0.75 g$.
Two forces act on the object: the tension $F_{T}$ upward and the downward pull of gravity $m g$. They are shown in the free-body diagram in Fig. $3-11(b) .$ As a rule begin your analysis with a diagram. Take up as the positive direction and write $+\Sigma F_{y}=m a$, in each case.

Notice that the tension in the cord is less than $m g$ in part $(d)$; only then can the object have a downward acceleration. Can you explain why $F_{T}=0$ if $a_{y}=-g$ ?

Key Concepts

Tension in a Cord

Tension is the pulling force transmitted along a string, rope, or cord, and it arises when the cord is used to support or accelerate an object. The tension must counteract other forces such as gravity and must be adjusted depending on whether the object is at rest, moving at constant velocity, or accelerating, thereby affecting the net force acting on the object.

Static Equilibrium vs. Accelerated Motion

Static equilibrium occurs when an object is at rest or moving at constant velocity, implying that the sum of all forces acting on the object is zero. In accelerated motion, however, a net force exists, and the balance between forces is disturbed, leading to acceleration. Analyzing these differences helps in understanding changes in tension under varying motion conditions.

Coordinate System and Sign Conventions

Choosing a consistent coordinate system with defined positive and negative directions is crucial for correctly applying vector arithmetic in force analysis. Sign conventions ensure that forces and accelerations are correctly accounted for in the mathematical equations derived from Newton's laws.

Newton's Second Law

Newton's second law states that the net force acting on an object is equal to its mass multiplied by its acceleration. This principle provides the mathematical framework for analyzing how forces influence the motion of an object and is central to solving problems involving acceleration and tension.

Free-Body Diagram

A free-body diagram is a visual tool used to map out all the forces acting on an object. It simplifies the analysis of the system by isolating the object and representing each force as a vector, indicating both its magnitude and direction. This tool is vital for setting up equations of motion using Newton's laws.

Transcript

00:02 Hi, in this problem we have an object of mass m and this object is supported by a chord.

00:12 So here, this object is only under the effect of the force of weight and the effect of the force of the tension in this court here.

00:24 So now we're given four different cases.

00:27 In the first case, the object is add trust and here, if we need to calculate forces, we need to know what is the acceleration.

00:38 And being addressed, this means that the acceleration is zero.

00:42 So in the first case, the acceleration, specifically in the y direction, is equal to zero.

00:50 From here, we know that the total or the sum of forces acting on an object is equal to the mass of this object multiplied by its acceleration.

01:01 So here, given that we count the positive direction to be upward and the negative direction to be downward.

01:12 So from here, the sum of forces is the force of tension minus the force of weight, and that's equal to the mass of the object multiplied by the acceleration.

01:25 In this first case, the acceleration is equal to zero.

01:28 So this term here is equal to zero and we have that the force of tension minus the force of weight is equal to zero from here we know that the force of tension is equal to the force of weight which is basically equal to the mass of the object multiplied by the gravitational acceleration and that's the answer to part a and that's the tension in the chord in that case now we will do the same for all different 4k so we have three remaining cases.

02:02 B, when the object is moving with constant velocity, when the object is moving with constant velocity, it's no big different from the object being addressed.

02:14 Why is this? because the acceleration is zero as well.

02:19 And given that the acceleration is zero, it's no different.

02:24 Speaking of forces, so here, let's apply the same that we did before, the acceleration, the y direction, is equal to 0.

02:32 So the sum of forces is equal to 0.

02:38 From here, we have that the force of tension should be equal to the force of weight, which is equal to mg...

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