Toque is related to rotator motion and can be defined as “a force that influences an object to rotate in the given axis.” This post is concerned with zero torque exerted on any object.
Zero torque refers to the torque acting on the rotational system being zero. It is the physical entity depending on the direction of the net force acting on the rotational axis and associated with the acceleration. In the below section, let us know more facts about zero torque.
Every object, irrespective of linear or rotational motion, attains an equilibrium state of motion at that time even though any external force acting on the object tends to equal zero. So if this condition is applicable for rotational motion, then we can see it as zero torque.
What is zero torque?
Zero torque is defined as “for any object under rotational motion, if net perpendicular force is zero, then rotational equilibrium sets up and thus angular acceleration is not possible; this condition is zero torque.”
The above state simply illustrates that if perpendicular force is zero, then a change in angular velocity does not occur, and angular momentum becomes constant. So that angular acceleration does not set up. The overall condition establishes an equilibrium state for the rotational motion. Thus, the object’s motion in a circular path becomes a uniform circular motion with constant velocity.
Can torque be zero?
Yes, torque can be zero. Suppose applied force on the object and its line of application crosses through the center of mass and if the rotational equilibrium condition balances all the forces. Under such conditions, the torque will be zero.
The above statement clarifies that if the net force exerted on the rotating object is parallel to the axis of rotation, the torque exerted on the object is nullified.
The torque of the stationary object is always zero because the net force acting on the object becomes zero by balancing all the other forces.
How can torque be zero?
Torque is the cross product of the radius of the circular path and force acting on the object moving in the circular path. It can be given as,
Its magnitude is given as
Where; τ is the torque, r is the radius of the circular path, |r| defines the position vector of the radius, F is the force and |F| defines the magnitude of the force, and θ is the angle.
When the force acting on the object under circular motion becomes zero, i.e., F=0, then
For a stationary object, there is no motion occurs, such that the radius will be zero, i.e., r=0; then
The torque is also defined as the product of rotational inertia and rotational acceleration. So the formula for torque is given as
τ=Iα, where I is the rotational inertia and α is the rotational acceleration.
From the above section, we know that their angular velocity and momentum become constant when the equilibrium is set up on the rotating body. Thus, rotational acceleration is not possible. Hence the equation can be rewritten as
In this case, I cannot be zero; it is constant for the equilibrium condition. Since α alpha is the second derivative of the speed, it shows that either its first derivative is zero or any constant value.
If the first derivative is zero, then we can say the body is in a stationary state. If the first derivative is constant, the body is under equilibrium with constant rotational velocity.
Under what conditions torque is zero?
In four conditions, the torque acting on the object becomes zero; they are;
- There is no force acting on the object.
- If the forces acting on the object are equal and opposite to each other, they can cancel each other.
- The force is acting on the rotational axis.
- If the force pointed towards the axis of rotation.
The above conditions simply determine the two cases at which the toque acting will be zero by considering the magnitude equation of torque τ=|r|×|F|sinθ
Case(i) at θ=0;
Since sin(0)=0, τ=0.
Case(ii) at θ=π
But sin(π)=0; hence τ=0.
The rotational equilibrium is equivalent to translational equilibrium; however, all the forces acting are nullified at this condition, and net torque will also become zero.
Where can torque be zero?
There are only two possibilities where torque can be zero; they are
- In stationary body
- A body under uniform circular motion.
For a stationary state, if there is no external force to push the body to trace the circular path, there is no torque.
A body under uniform circular motion refers to moving in a circular orbit with constant speed. The angular acceleration of the body is not possible. Hence the torque will be x=zero because angular acceleration and torque are interconnected.
Even if the torque is absent, the body still manages to rotate in a uniform circular orbit due to the radial force exerted on the body. The radial velocity does not become constant and gives rise to exert radial acceleration, which results in uniform rotation of the body.
What happens when net torque is zero?
When the net torque is zero, the angular momentum of the rotating system is conserved, and angular acceleration will be zero. The rotational equilibrium is established in the rotating system by keeping the angular velocity constant.
Basically, torque is the derivative of angular momentum, it can be written as,
If τ=0, then
Integrating on both side, we get
Can there be torque when net force is zero?
No, even though the net force acting on the rod is zero. The situation looks converse to the theories of physics, but there will be a torque due to radial acceleration.
For example, consider a rod at rest in a vacuum, and no external force is acting on the rod. If upward force is exerted at the center of the rod, the rod begins to accelerate, but it does not rotate because force is applied to the center.
Now, if downward force is exerted on the rod at one of its ends, the rod accelerates and begins to rotate because of the distance between the applied downward force and the center of the rod.
If you calculate the net force acting on the rod, it will be zero. Because the magnitude of the upward and downward forces is the same and exertion is the opposite, they cancel out. At this condition, the center of mass of the rod does not accelerate; however, the torque exerted on the rod manages the rod to rotate.
In short, we can summarize the above example as the net force acting on the object is essential to explain the acceleration of any object at its center of mass, but it does not explain the orientation. And torque can also be exerted even if the net force is zero.
Can torque be zero when force is nonzero?
The torque will be zero in some situations, but some forces are still exerted, which do not vanish and remain nonzero. In such conditions, the body’s motion is linear, not rotational.
For example, consider pushing a heavy block. For this action, a huge amount of force is required. The block does not rotate even if you apply such a huge force to it. Thus, torque remains zero even if some external force influences the block to move.
Can a single force produce a zero torque?
At the center of gravity, a single force can produce zero torque, but a linear force accompanies the torque. There is no opposing force exerted at the center of gravity. If there is any external constraint, it will end up with a linear uniform accelerated body with zero torque.
In an unconstraint system like a rigid body, a single force generates the non-steady motion and inertia, so the force at the center of gravity tends to rotate the body, and either positive or negative torque will be generated. Thus, a constraint system must produce zero torque from a single force.
Zero torque uses
Some advanced technologies, such as robotics, use the zero torque concept. Some of such uses of zero torque are listed below:
- Switched reluctance motors
- Pre-buckled beams
- Exoskeleton mechanism
- PMDFI generators
- EV coasting technology
- Spinning gyroscope
Switched reluctance motors
Switched reluctance motors are electric motors that run by the principle of reluctance torque. The performance of SRM is different from the ordinary DC motor; they consist of winding to deliver the power in the stator keeping the zero torque.
Zero torque is widely used in the mechanism of pre-buckled beams, in which a balanced statistical mechanism is employed with a single rotational degree of freedom. The statistical balance is done by merging positive and negative stiffness in an annular domain.
For safety purposes, the exoskeleton of the human body implements a zero-torque controller so that exoskeleton can easily respond to the human physical action.
The mathematical modeling of the permanent magnet doubly fed-up induction generator is based on the rotating coordinates in which zero torque is enhanced to maintain the low voltage ride.
EV coasting technology
Induction coupling rotators are used to prevent undesirable braking in electric vehicles, which balances the zero torque.
A freely spinning gyroscope like a bicycle wheel utilizes zero torque rotation and force exerted from the constant angular momentum.