Physics -- Rotational Equilibrium and Dynamics

Standard Definition of Torque

r x F

Center of Mass

The point at which the mass of a body or system is concentrated and about which all parts balance.

Center of Mass Equation

x_cg=(sum(m_ix_i)/sum(m_i)); y_cg=(sum(m_iy_i)/sum(m_i))

Mechanical Equilibrium

An object that has a net force of 0 and a net torque of 0.

Standard Moment of Inertia

I=sum(mr²)

Rotational Second Law of Motion

(sum)t=Ia

Rotational Kinetic Energy

KE_r=1/2Iw²

Angular Momentum

L=Iw

Rotational Work

𝑛𝑒𝑡𝑊=(𝑛𝑒𝑡𝜏)𝜃

Rotational Impulse

(𝑛𝑒𝑡𝜏)t

Why can’t you put your heels firmly against a wall and then bend over without falling?

Your center of mass must remain under your feet to avoid falling and when you bend over, your center of mass shifts outside your base of support, causing you to lose balance.

If you see an object rotating, is there necessarily a net torque acting on it?

No, the object can be rotating without accelerating angularly.

Why does a long pole help a tightrope walker stay balanced?

The pole increases the tightrope walker’s moment of inertia, helping the tightrope walker resist angular acceleration.

A tape recorder is pulled at a constant speed by a motor. As the tape is pulled off, the radius of the roll of remaining tape decreases. How does the torque on the reel change with time? Is the reel more likely to snap nearly full or nearly empty?

As the radius of the roll decreases, the torque on the reel also decreases because torque is directly proportional to the radius (torque = radius x force). The reel is more likely to snap when it is nearly full, as the greater radius creates more torque and stress on the reel.

Give an example in which the net force on an object is zero but the net torque is nonzero.

An example would be a see-saw where two equal weights are placed at different distances from the pivot point. The net force is zero as the weights balance each other, but the torque can be nonzero if the weights are at different distances from the pivot.

Give an example in which the net torque acting on an object is zero but the net force is nonzero.

An example would be a person pushing a stationary car. The net force is nonzero due to the push, but if the force is applied at the center of mass, the net torque is zero, resulting in no rotational movement.