Solving Net Torque Problems

G11-1. Solving Net Torque Problems

Solving a net torque problem is similar to solving a net force problem, but there are some differences. For both types of problems, you have to find the sum of something, either forces or torques. For net force problems, though, you set F_net = ma, while for net torque problems, you set τ_net = Iα. Here are some other differences:

For net torque problems, you must select an axis of rotation about which to calculate moment arms and torques.
Net force problems are vector problems. You solve for force components along each axis. For net torque problems, you have to indicate whether a torque tends to produce clockwise or counterclockwise rotation.

We start with a method for calculating net torque and then show how to solve for the angular acceleration.

Calculating Net Torque

Here are important things to do in calculating net torque.

Select an axis of rotation. Show on a diagram the axis that you select.
Draw the forces extending from their points of application. (While this wasn't important for net force problems, it is important for net torque problems.)
Draw position vectors extending from the axis of rotation to the points of application of each of the forces.
For each force, identify the angle, θ, from the position vector to the force. In order to determine the correct angle, imagine r and F starting from the same point and curl the fingers of your right hand from the position vector to the force vector. The angle between the two vectors is the one you want. The vector may be acute or obtuse. Be sure to identify it correctly, because the sine of the angle is used in the calculation of the moment arm.
Construct the moment arm, , of each force. This may require extending the line of application of the force.
Determine the length of each moment arm using .
Write the net torque equation, . Substitute the magnitude of each torque using . Give each torque the correct sign according to the convention that torques tending to produce counterclockwise are +.
Substitute given values and solve for the net torque. The sign of the result will tell you whether the angular acceleration due to the net torque is clockwise or counterclockwise.

Example. A circular disc is rotated about an axis O through its center by the application of two forces. A force of magnitude 11 N is exerted at a distance of 0.34 m from the axis and at an angle of 58° from a radial line extending from the axis through the point of application Q of the force. A second force of magnitude 15 N is exerted at a distance of 0.26 m from the axis and at an angle of 119° from a radial line extending from the axis through the point of application P of the force. Determine the net torque on the disc about its center and which way the net torque accelerates the disc.

Given:
F₁ = 11 N
F₂ = 15 N
r₁ = 0.34 m
r₂ = 0.26 m
θ₁ = 58°
θ₂ = 119°
Goal:
Find τ_net and the direction of the angular acceleration
1. The axis has already been chosen for us at the center of the disc.
2. The forces are drawn extending from points P and Q.
3. The position vectors are drawn from point O to points P and Q.
4. The angle through which the position vector rotates into the force vector is indicated for each force. See the supplementary diagrams to the right for how the angles are determined.
angle 1 angle 2

We'll now add some auxiliary lines and angles in the next diagram. (Two diagrams wouldn't normally be needed, but we're providing a second one for clarity.)

Calculating moment arms

5. The line of application of F₁ has been extended so that the moment arm of the force can be drawn perpendicular to the line of application. The moment arm of F₂ is also drawn.

6. The moment arm of F₁ is opposite the angle α, which is equal to θ₁. The moment arm of F₂ is opposite the angle β, which is supplementary to θ₂.

7. The net torque equation is written. The torque due to F₁ (F₂) is negative (positive), because it tends to produce clockwise (counterclockwise) rotation. Note that the symbols τ₁ and τ₂ represent the magnitudes of the torques. The signs are added explicitly. The expressions obtained above for moment arm are substituted.
8. Values are substituted and the value of the net torque is solved for. The result is positive, indicating that the angular acceleration is counterclockwise. Note that while the units are Nm, we don't call them joules. A joule is used to represent a unit of work or energy. Torque is not energy.

Calculating Angular Acceleration

Once the net torque is known, one calculates the angular acceleration using α = τ_net/I, where I is the moment of inertia of the rotating, rigid body. We provide the following complete, example solution.

The situation is that of a thin rod of uniform density hinged to a level table at one end. See Figure 1 below. The mass of the rod is M and its length is L. Now consider Figure 2. We select the hinge at point A for the axis of rotation. Note that we don't have to select the physical axis of rotation as the point about which to calculate torques. However, it's convenient to do so in this situation. That's because the hinge exerts forces on the rod both to support it and to keep it from sliding horizontally. Since these forces act through the axis of rotation, they have 0 moment arms and hence exert 0 torque. Thus, the only force that exerts a torque is the weight, Mg, of the rod. We take this force as acting vertically from the center of mass. The moment arm of this force is the perpendicular distance, AB, from the hinge to the line of action of the force. This distance is AB = rcosθ, where r is the distance from the hinge to center of mass. You may wonder why we use the cosine function here rather than the sine function. The reason has to do with the way the angle θ is defined. With the angle shown in the diagram, the perpendicular distance from the axis of rotation to the line of application of the force is the side of the right triangle adjacent to θ. Thus, we use the cosine.

We can now write an expression for the torque produced by the weight of the rod. Since the rod is of uniform density, we can say that the center of mass is at the geometric center of the rod, thus, r = L/2 and AB = (L/2)cosθ. The torque exerted by the weight of the rod is therefore τ_Mg = -Mg(L/2)cosθ. The minus sign indicates that the torque produces a clockwise rotation.

Figure 1

Figure 2

hinged rod

Since the weight is the only force exerting a torque on the rod, then τ_net = τ_Mg = -Mg(L/2)cosθ. We can now solve for the angular acceleration.

α = τ_net/I

= -Mg(L/2)cosθ / I

= -Mg(L/2)cosθ / (ML²/3)

= -(3/2)(g/L)cosθ

We substituted the expression from Table 10-1 of the text for the moment of inertia of a thin rod with axis at one end. Now let's do some checks. We've already addressed the sign. The units on the right are (m/s²)/m = 1/s², which are the units of angular acceleration. Consider special values of the angle next. When θ = 90^o, α = 0. This makes sense, because the rod would be vertical and in equilibrium. (Note that the equilibrium is unstable, because just a small motion would allow it to start rotating.) When θ = 0^o, α = -(3/2)(g/L). The angular acceleration is greatest just before the rod hits the table. That makes sense, because the moment arm of the weight is greatest at that point. But does a magnitude of (3/2)(g/L) make sense? Consider the tangential acceleration at that point. This is vertically downward and given in magnitude by |a_t| = L|α| = (3/2)g. This says that the acceleration of the end of the board is 50% greater than g. Does it make sense that the acceleration of a falling object can be greater than g? The answer is yes, assuming that the object is part of a rotating, rigid object as in this situation. There are other points of the object with tangential accelerations less than g. For example, the center of mass has |a_CM| = (L/2)|α| = (3/4)g or 75% of g. This suggests that there's a point on the rod that has a tangential acceleration of g. We leave it as an exercise for you to determine where that point is.