Rotational Mechanics

Torque is “turning force” T = F r or Force times the lever arm radius (also may be written T = F d or T = F L)

For Maximum Torque… You must apply the force at a RIGHT ANGLE to the lever arm (any other angle lessens the torque) The true formula is Τ = F r (sin θ ) Since the sin of a 90 degree θ is _____ , this all makes perfect sense ;-)

If you double the lever arm distance, you________ the Torque The English unit for torque is the Foot-Pound (ft lb) The metric unit for torque is the Newton-meter (N m)

One child weighs 400 N and sits 1 One child weighs 400 N and sits 1.5 meters from the fulcrum (pivot point or center) of the see saw. How much torque does he produce? His little sister weighs only 300 N. How far must she sit from the fulcrum to be balanced?

SOLUTIONS T = F d or F r or F l 1) T = 400 N (1.5 m) = 600 N m of Torque 2) T = T so Fd = Fd 400 N (1.5 m) = 300 N (x) 600 N m = 300 N (x) 600 N m = x = 2 m 300 N Link to Torque

All torques are CLOCKWISE or COUNTERCLOCKWISE

What are ways you can INCREASE TORQUE?

ROTATIONAL INERTIA Once something is spinning, it naturally ______________________.

Rotational inertia basic formula: I = m r 2 Formula varies for different shapes and distributions of mass! The bigger the radius, the HIGHER the rotational inertia The more MASS is AWAY FROM the axis of rotation, the HIGHER the rotational inertia

What would have a higher rotational inertia, a regular basketball filled with air, or a basketball filled with foam? Which would accelerate faster (or win a race down a short ramp)? Which would tend to keep rolling longer (or win a race down a long ramp or distance?) Which is easier to start spinning? Which is easier to stop spinning? Which tends to keep spinning longer?

Example: Assuming the earth is a solid sphere rotating around an axis through its poles, the rotational inertia of the earth would be: I = 2/5 mr2 = 2/5 (5.98 x 1024 kg) (6.37 x 106 m)2 = 9.71 x 1037 kg m Link

Why do you bend your legs when you run? Does it take more torque to move long legs, or short legs? Does it take more torque to move bent legs or straight legs?

Does it take more torque to turn big wheels/tires or small? Why? Would you get better gas mileage around town with big wheels/tires or small? Why? Would you get better gas mileage on a long deserted highway trip through flat country with big wheels/tires or small? Why?

Why do you hold your arms out when walking across a log or balance beam?

How does holding a pole help? Do the buckets make the pole have a higher rotational inertia, or lower? Does the pole tend to twist (get off balance) more easily if buckets are empty or full?

To INCREASE his rate of spin, what does the skater do? Does this INCREASE or DECREASE his rotational inertia?

If a diver, gymnast, etc., goes from an extended to a tucked (balled up) position: Does their rotational inertia INCREASE, DECREASE, OR STAY THE SAME? Does their angular velocity INCREASE, DECREASE, OR STAY THE SAME? Does their angular momentum INCREASE, DECREASE, OR STAY THE SAME?

Remember that linear momentum is (p = mv) Angular momentum is (L = mvr or L = Iω) Angular momentum is ALWAYS CONSERVED, just like linear momentum Iω = Iω If rotational inertia increases, what happens to rotational speed?