Torque

Torque is the underlying force that governs rotation and is useful for describing the operation of both simple and complex machines.

Much like how a net force causes linear acceleration a in a translational system, a net torque, typically represented by the Greek letter t, is a force that causes angular acceleration a in a rotational system.

However, if multiple torques acting on a system are made to balance one another, then the net torque will be zero, and the system will be in equilibrium.

The goal of this video is to understand the components of torque by placing weights at different positions on a freely rotating beam to achieve rotational equilibrium.

Before using weights to balance a beam, let's revisit the concepts of torque and rotational equilibrium. A good example of torque is when you have a flat tire and have to use a wrench to loosen a nut before you can change it.

Torque is defined as the cross product of the distance, r, from the axis of rotation at which the force is applied and the force. This distance is also called the lever arm. Note that only the perpendicular component of the force, found using the sin of the angle theta between the force and the rotation arm, contributes to the magnitude of the torque.

It is evident from the equation that by moving the applied force from the middle of the wrench to the end, you double the lever arm and thus double the torque being used to loosen the nut. If the nut still won't budge, you might need to figure out how to increase the perpendicular force.

Now consider another system, where a weight of mass m is attached to a beam that can rotate. Knowing the relationship between linear and angular acceleration, and multiplying both sides of the equation by r, gives a new definition for torque. Now, mr2 is nothing but the system's rotational inertia I and this equation of torque represents the rotational equivalent of Newton's second law, where a torque can cause angular acceleration. Please watch JoVE's Science Education video on Rotational Inertia for further information on this subject.

Now, if the beam is leveled and the weight is removed, there is no net torque on the system and so the angular acceleration must also be zero. Therefore, the system at rest will not rotate and is said to be in rotational equilibrium. For more information on this concept, please watch the video on Equilibrium and Free-Body Diagrams.

Rotation equilibrium can also be established by properly positioning weights on opposite sides of the axis of rotation, so that they equally oppose one another. Conventionally, with respect to the axis of rotation, torque is positive for counter-clockwise rotation and negative for clockwise rotation.

Now that you understand how torque can affect a rotational system, let's see how to apply these forces to achieve equilibrium. This experiment consists of a beam with equally spaced hooks for attaching weights, a protractor, a force scale, and numerous weights with 100 g and 200 g masses.

Initially, two weights are used to establish rotational equilibrium with a 200 g weight connected to the first hook on the right. Connecting another 200 g weight to the first hook on the left should prevent the beam from rotating. Remove the weight from the left side and place a 100 g weight in the proper position to balance the torque from the right side.

Next, three weights are used to balance the torque starting with 100 g weights on both the first and third hooks on the right. Properly position a 200 g weight on the left side so that the net torque on the system is zero. Next, remove the weight and use a 100 g weight to re-establish equilibrium.

Subsequently, multiple weights are employed to balance the beam with a 200 g weight connected to the fourth hook on the right. Using any combination of 100 g and 200 g weights, determine three configurations on the left-hand side that can achieve rotational equilibrium.

Next, with the 200 g weight still connected to the fourth hook on the right, calculate the force required to balance the torque for each of the hooks on the left. Attach the force scale to the first hook on the left, making sure it is perpendicular to the beam, and pull it down until the beam is level and record force value. Repeat this procedure for each hook on the left.

Lastly, with the 200 g weight still attached, connect the force scale to the third hook on the left and level the beam. And, using a protractor, allow the beam to rotate to the right by 30 degrees. Making sure the force scale is perpendicular to the beam, record the force value. Increase the rotation angle to 60 degrees and record this force value.

Each of the balanced beam experiments confirms that a proper configuration of weights can establish equilibrium where the net torque is zero. No net torque implies that no angular acceleration occurs and therefore the beam does not rotate if released from rest. This rotational equilibrium is particularly evident with the six different configurations of 100 and 200 g weights on the left side, which can balance the 200 g weight attached to the right outermost hook.

In the next experiment, the force scale allowed for a more continuous measurement of the torque required for equilibrium. Since the force scale is perpendicular to the beam, just like the weight, the force FL at equilibrium could be calculated using this formula. And this table shows the calculated force for different hooks on the left side with a constant 200g weight on the outmost hook on the right side.

When the beam is rotated from the horizontal by an angle theta, only a component of the gravitational weight, given by this formula, is contributing to the torque. Consequently, the measured force will be less than the value observed for the level beam and will decrease with increasing angle.

The basic principles of torque can be invaluable when trying to understand rotating mechanical systems and how this can translate to linear motion.

A seesaw perfectly demonstrates torque with people generating force on either side of the fulcrum to create rotation. When both sets of people have similar lever arms, the heavier set of people will generate more torque and the other set of people will be lifted up. Conversely, in order to raise the heavier set of people up, they must reduce their moment arm by sliding toward the fulcrum.

A vehicle's torque plays a significant role in its performance, as evident from Newton's second law of angular acceleration. For vehicles with the same inertia, greater torque generates greater angular acceleration, which is directly proportional to the vehicle's linear acceleration. Similarly, if two vehicles have the same acceleration, increased torque would accommodate more inertia and therefore allow a vehicle to tow a massive load.

You've just watched JoVE's introduction to Torque. You should now understand the principles of torque and how it can be used to establish rotational equilibrium or generate angular acceleration. Thanks for watching!