Question

Torque and Equilibrium Exercise Questions

1) Describe the conditions necessary for equilibrium on a balance.

2)  When the pivot point of a balance is not at the center of mass of the balance, how is the net torque on the balance calculated? When a force is applied directly to the pivot point of a balance, what is the torque due to that force?

3) To maintain the same amount of torque due to a mass on a balance as the mass is increased, how should the position of the mass change?

4) Why is the handle on a door far from the hinge? Use the definition of torque in your answer.

Answer 1

Ans 1: CONDITIONS NECESSARY FOR EQUILIRIUM ARE:

An object is in equilibrium if ;

1. The resultant force acting on the object is zero.
2. The sum of the moments acting on an object must be zero.

First Condition of Equilibrium:

For an object to be in equilibrium, it must be experiencing no acceleration. This means that both the net force and the net torque on the object must be zero. Here we will discuss the first condition, that of zero net force.

In the form of an equation, this first condition is:

F_net=0

In order to achieve this conditon, the forces acting along each axis of motion must sum to zero. For example, the net external forces along the typical x– and y-axes are zero. This is written as

net F_x=0 and net F_y=0

The condition Fnet=0Fnet=0 must be true for both static equilibrium, where the object’s velocity is zero, and dynamic equilibrium, where the object is moving at a constant velocity.

Below, the motionless person is in static equilibrium. The forces acting on him add up to zero. Both forces are vertical in this case.

Person in Static Equilibrium: This motionless person is in static equilibrium.

Below, the car is in dynamic equilibrium because it is moving at constant velocity. There are horizontal and vertical forces, but the net external force in any direction is zero. The applied force between the tires and the road is balanced by air friction, and the weight of the car is supported by the normal forces, here shown to be equal for all four tires.

A Car in Dynamic Equilibrium: This car is in dynamic equilibrium because it is moving at constant velocity. The forces in all directions are balanced.

2nd CONDITION OF EQUILIBRIUM:

The second condition of static equilibrium says that the net torque acting on the object must be zero.

A child’s seesaw, shown in, is an example of static equilibrium. An object in static equilibrium is one that has no acceleration in any direction. While there might be motion, such motion is constant.

Two children on a seesaw: The system is in static equilibrium, showing no acceleration in any direction.

If a given object is in static equilibrium, both the net force and the net torque on the object must be zero. Let’s break this down:

Net Force Must Be Zero:

The net force acting on the object must be zero. Therefore all forces balance in each direction. For example, a car moving along a highway at a constant speed is in equilibrium, as it is not accelerating in any forward or vertical direction. Mathematically, this is stated as F_net = ma = 0.

Net Torque Must Be Zero:

The second condition necessary to achieve equilibrium involves avoiding accelerated rotation (maintaining a constant angular velocity ). A rotating body or system can be in equilibrium if its rate of rotation is constant and remains unchanged by the forces acting on it.

Hence we can say that:

Condition 1: ∑Fx=0,∑Fy=0, translational equilibrium

Condition 2: ∑τ=0, rotational equilibrium

Ans2:  Torque is the rotational equivalent of a force in producing a rotation and is defined to be

=rFsinθ

where τ is torque, r, is the distance from the pivot point to the point where the force is applied, F is the magnitude of the force, and θ is the angle between F and the vector directed from the point where the force acts to the pivot point. The perpendicular lever arm r_⊥ is defined to be

r⊥=rsinθ

so that

=r⊥F

For example:

A wrench produces a torque on a nut if a force is applied to it correctly (see Figure 1). The equation for torque is:

=rFsinθ

Figure 1. Variables of the torque equation shown for a wrench and nut. The nut’s center is the pivot point.

To produce a torque, the force F must be applied at some distance rrr away from the pivot point. Since only the perpendicular component F⊥,produces torque, the equation includes sin

Figure 2. Components of the applied force F aligned to the lever arm.The perpendicular component is F⊥, and the parallel component is F∥​

The magnitude of the torque depends on:

• Applied force F: Larger forces increase torque.

• Radius r: Increasing the radius increases the torque.

• Angle between the force and lever arm : Directing a force perpendicular to the lever arm increases the torque.

An applied force can result in zero torque if there is no lever arm or the applied force is parallel to the lever arm (see Figure 3 and 4 below).

Figure 3. Lever arm: these applied forces result in no torque on the wrench because of no lever arm r.

Figure 4. Direction of force: these applied forces result in no torque on the wrench because the applied force is parallel to the lever arm.

Ans 3: Rotational Inertia or Moment of Inertia:

The rotational equivalence of mass is moment of inertial, I. It accounts for how the mass of an extended object is distributed relative to the axis of rotation. For a point mass m connected to the axis of rotation by a massless rod with length r, I = mr².

If the mass is distributed at different distances from the rotation axis, then there will be different values of moment of Inertia.

Torque causes rotational motion with angular (or rotational) acceleration α.

τ_net = Iα

where I is the moment of inertia of the system and α is the angular acceleration. This equation is the angular equivalent of Newton's second law:

F_net = ma

When the net torque is zero, the object will not change its state of rotational motion—i.e., it will not start rotating or stop rotating or change the direction of its rotation. It is said to be in rotational equilibrium. If the sum of the forces acting on the object is also zero, the object is in translational equilibrium and will not change its state of translational motion, that is, it will not speed up or slow down or change its direction of motion. Whenever both of these conditions

	τ = 0

and

	F = 0

are met, the object is said to be in static equilibrium.

Ans 4: Usually the door knob is placed far away from the fulcrum (where the door is attached to the wall with hinges) as shown :

Illustration 1. ◦ If you pull on the knob to open the door, you have a pretty easy time because the force you are using causes a lot of torque (twisting) around the fulcrum (hinges) to open the door.

• Imagine that the door knob is now placed near the centre of the door, closer to the hinge, as shown :

Illustration 2. ◦ If you pull on the door knob with the same force as before, it will be more difficult to open the door. ◦ This is because the torque (twisting force) has decreased

. The formula to measure the amount of torque is a short one, but applying it depends on the question being asked. =F ℓ

τ = torque (N∙m)

F = force (N)

ℓ = lever arm distance (m) •

At the end of a calculation of torque, it is common to give the torque a positive or negative value, depending on the direction it is turning.

◦ Clockwise (CW) torque = negative

◦ Counterclockwise (CCW) torque = positive

• Here the force was kept constant but the lever arm distance was changed. This is why we did not get as much torque in the second situation.

Hence,

The reason is because of torque which basically creates a rotational effect.
=rXF with requisite vector notations.
Torque is expressed as the cross product of distance from hinge/centre of mass (whichever is the chosen axis - in this case, it is the hinge) and the force applied. As distance increases, torque increases and so the rotational effect increases.
Hence, it is easier to push open a door by holding the knob than by pushing closer to the hinge.