Torque
Torque (τ) is a force that causes an object to turn. If you think about using a wrench to tighten a bolt, the closer
to the bolt you apply the force, the harder it is to turn the wrench, while the farther from the bolt you apply the
force, the easier it is to turn the wrench. This is because you generate a larger torque when you apply a force at
a greater distance from the axis of rotation.
Let’s take a look at the example of a wrench turning a bolt. A force is applied at a distance from the axis of
rotation. Call this distance r. When you apply forces at 90 degrees to the imaginary line leading from the axis of
rotation to the point where the force is applied (known as the line of action), you obtain maximum torque. As
the angle at which the force applied decreases (θ), so does the torque causing the bolt to turn. Therefore, you
can calculate the torque applied as:
In some cases, physicists will refer to the rsinθ as the lever arm, or moment arm, of the system. The lever arm is
the perpendicular distance from the axis of rotation to the point where the force is applied. Alternately, you
could think of torque as the component of the force perpendicular to the lever multiplied by the distance r. Units
of torque are the units of force × distance, or Newton-meters (N·m).
Objects which have no rotational acceleration, or a net torque of zero, are said be in rotational equilibrium. This implies
that any net positive (counter-clockwise) torque is balanced by an equal net negative (clockwise) torque.
Moment of Inertia
Previously, the inertial mass of an object (its translational inertia) was defined as that object’s ability to resist a
linear acceleration. Similarly, an object’s rotational inertia, or moment of inertia, describes an object’s
resistance to a rotational acceleration. The symbol for an object’s moment of inertia is I.
Objects that have most of their mass near their axis of rotation have a small rotational inertia, while objects that
have more mass farther from the axis of rotation have larger rotational inertias.
For common objects, you can look up the formula for their moment of inertia. For more complex objects, the
moment of inertia can be calculated by taking the sum of all the individual particles of mass making up the
object multiplied by the square of their radius from the axis of rotation. This can be quite cumbersome using
algebra, and is therefore typically left to calculus-based courses or numerical approximation using computing
systems.
Newton's 2nd Law for Rotation
In the chapter on dynamics, you learned about forces causing objects to accelerate. The larger the net force, the
greater the linear (or translational) acceleration, and the larger the mass of the object, the smaller the
translational acceleration.
The rotational equivalent of this law, Newton’s 2nd Law for Rotation, relates the torque on an object to its
resulting angular acceleration. The larger the net torque, the greater the rotational acceleration, and the larger
the rotational inertia, the smaller the rotational acceleration:
SAMPLE SOLUTIONS
Question: A pirate captain takes the helm and turns the wheel
of his ship by applying a force of 20 Newtons to a wheel
spoke. If he applies the force at a radius of 0.2 meters from
the axis of rotation, at an angle of 80° to the line of action,
what torque does he apply to the wheel?
Answer:
Question: A mechanic tightens the lugs on a tire by applying a
torque of 110 N·m at an angle of 90° to the line of action. What
force is applied if the wrench is 0.4 meters long?
Answer:
Question: How long must the wrench be if the mechanic is only capable of applying a force of
200N?
Answer:
Question: Calculate the moment of inertia for a solid
sphere with a mass of 10 kg and a radius of 0.2 m.
Answer:
Question: Calculate the moment of inertia for a hollow
sphere with a mass of 10 kg and a radius of 0.2 m.
Answer:
Question: What is the angular acceleration experienced by a
uniform solid disc of mass 2 kg and radius 0.1 m when a net
torque of 10 N·m is applied? Assume the disc spins about its
center.
Answer:
PRACTICE PROBLEMS:
Use the samples for guidance -SHOW WORK AND FORMULAS for all problems
1.
Question: You are on wheel of fortune, you spin the giant wheel by applying 74N of force to the wheels
spoke. If he applies the force at a radius of 2.3 meters from the axis of rotation, at an angle of 75° to the
line of action, what torque does he apply to the wheel?
2a, A monster truck mechanic tightens the lugs on a tire by applying a torque of 545 N·m at an angle of 90°
to the line of action. What force is applied if the wrench is 2.3 meters long?
2.b. How long must the wrench be if the mechanic is only capable of applying a force of 400N?
3. While hiking in the winter a snow ball with a solid mass of 422 Kg and a radius of 6 m is set to roll down a
hill, what is the moment of inertia that must be overcome for the ball to begin rolling down the hill.
4. What is the angular acceleration experienced by a (penny) uniform solid disc of mass .5 kg and radius 0.15 m when a
net torque of 5 N·m is applied? Assume the disc spins about its center.