Use this law just as you did in the Newton's Laws lessons. First choose a pivot point to take **all** torques around, and then add up all the torques acting on an object and that will equal the moment of inertia multiplied by the angular acceleration.

**Torques** produce angular accelerations, but just as masses resist acceleration (due to inertia), there is an inertia that opposes angular acceleration. The measure of this inertial resistance depends on the mass, but more importantly on the distribution of the mass in a given object. **The** *moment of inertia,***is the rotational version of mass.** Values for the moment of inertia of common objects are given above. Torques have only two directions: those that produce clockwise (CW) and those that produce counterclockwise (CCW) rotations. The angular acceleration or change in

Many separate torques can be applied to an object. The angular acceleration produced is

Angular accelerations are produced by net torques,with inertia opposing acceleration; because

This is the rotational analog of Newton's 2nd law

The net torque is the vector sum of all the torques acting on the object. When adding torques it is necessary to subtract CW from CCW torques.

#### Example

Some old doors have the door knob in the center of the door like in the picture below. If you had a door of mass 20 kg and 1.5 m wide, what would be the angular acceleration if you pushed at the center of the door with a force of 50 N? What would the angular acceleration be if you pushed at the far end of the door with the same force? Where should you push the door if you want to open it the fastest?

To solve this problem, we will plug in the known values into Newton's second law for rotation and solve for

Now we'll calculate the angular acceleration when pushing from the end of the door.

Clearly, it is much faster and easier to open doors when pushing from the point farthest from the hinge.

### Review

- In the figure we have a horizontal beam of length,
L , pivoted on one end and supporting2000N on the other. Find the tension in the supporting cable, which is at the same point at the weight and is at an angle of30 degrees to the vertical. Ignore the weight of the beam. - Two painters are on the fourth floor of a Victorian house on a scaffold, which weighs
400N . The scaffold is3.00m long, supported by two ropes, each located0.20m from the end of the scaffold. The first painter of mass75kg is standing at the center; the second of mass,65.0kg , is standing1.00m from one end.- Draw a free body diagram, showing all forces and all torques. (Pick one of the ropes as a pivot point.)
- Calculate the tension in the two ropes.
- Calculate the moment of inertia for rotation around the pivot point, which is supported by the rope with the least tension. (This will be a compound moment of inertia made of three components.)
- Calculate the instantaneous angular acceleration assuming the rope of greatest tension breaks.

- A horizontal beam of weight
60N and1.4m in length has a100N weight on the end. It is supported by a cable, which is connected to the horizontal beam at an angle of37 degrees at1.0m from the wall. Further support is provided by the wall hinge, which exerts a force of unknown direction, but which has a vertical (friction) component and a horizontal (normal) component.- Find the tension in the cable.
- Find the two components of the force on the hinge (magnitude and direction).
- Find the coefficient of friction of wall and hinge.

- On a busy intersection a
3.00m beam of150N is connected to a post at an angle upwards of20.0 degrees to the horizontal. From the beam straight down hang a200N sign1.00m from the post and a500N signal light at the end of the beam. The beam is supported by a cable, which connects to the beam2.00m from the post at an angle of45.0 degrees measured from the beam; also by the hinge to the post, which has horizontal and vertical components of unknown direction.- Find the tension in the cable.
- Find the magnitude and direction of the horizontal and vertical forces on the hinge.
- Find the total moment of inertia around the hinge as the axis.
- Find the instantaneous angular acceleration of the beam if the cable were to break.

- The medieval catapult consists of a
200kg beam with a heavy ballast at one end and a projectile of75.0kg at the other end. The pivot is located0.5m from the ballast and a force with a downward component of550N is applied by prisoners to keep it steady until the commander gives the word to release it. The beam is4.00m long and the force is applied0.900m from the projectile end. Consider the situation when the beam is perfectly horizontal.- Draw a free-body diagram labeling all torques.
- Find the mass of the ballast.
- Find the force on the vertical support.
- How would the angular acceleration change as the beam moves from the horizontal to the vertical position. (Give a qualitative explanation.)
- In order to maximize range at what angle should the projectile be released?
- What additional information and/or calculation would have to be done to determine the range of the projectile?

**Review (Answers)**

2300N - b.
771N,1030N c.554kgm2 d.4.81rad/sec2 - a.
300N b.240N,−22N c..092 - a.
2280N b.856n toward beam,106N down c.480kgm2 d. \begin{align*}3.8 \;\mathrm{rad/sec}^2\end{align*} - b. \begin{align*}345 \;\mathrm{kg} \end{align*} using \begin{align*} 9.8 \;\mathrm{m/s}^2\end{align*} for acceleration of gravity c. 4690 N