Tan Pushing Limits

Tan Pushing Limits

More physics help (Friction)?

A block of mass m rests on a rough horizontal surface. Coefficient of friction is u. A pulling force F = mg acts at angle θ with the vertical slide of the block. Find when the block can be pulled along the floor.
(a) Cot θ ≥ u
(b) tan θ ≥ u

What are the limits of cot θ in terms of u if the force is pushing rather than pulling?

If θ is smaller than a certain angle α, the block cannot be moved no matter what the force. Find α in terms of arccot u

F makes θ with vertical. Therefore, horizontal component of F = Fsinθ and vertical component = Fcosθ
After breaking F into these two components, we find that the forces on the block are
1. Fsinθ forward = mgsinθ forward
2. Fcosθ upward = mgcosθ upward (since F = mg)
3. Friction f backward
4. Normal force N upward
5. Weight mg downward

There is no acceleration in vertical direction. Therefore, total force in vertical direction = 0
Or, mgcosθ + N-mg = 0
Or, N = mg-mgcosθ
Or, N = mg(1-cosθ)
f = uN = umg(1-cosθ)
Block can be pulled when horizontal component of F exceeds friction.
Or, mgsinθ > umg(1-cosθ)
Divide by mg
sinθ > u(1-cosθ)

Do we have to choose between a and b? Or are a and b two different questions?

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Stomach Exercises: Aquasize to a Flatter Stomach!

When the weather is warm, there is nothing quite like going for a refreshing swim at the local pool. To add to the benefits, you can use it as an opportunity to work on those stomach muscles. The water provides natural resistance, making the pool a great place to do stomach exercises to flatten out that tummy. Despite the greater resistance water has than air, aquasizing is low-impact, meaning less strain on joints. Try out these exercises next time you take a trip down to your pool. Remember, consult your doctor before beginning a new workout routine, and always warm up properly to prevent injury.

The first set of exercises to try is called the jump and dig. There are two moves in this set, one for the upper body and one for the lower body. The former is great for your obliques, and the latter works on your abdominals. To start, stand in water that is between your belly button and chest. To work the lower body (including your abdominals), put your feet a wide distance apart, then jump so your knees come up to the surface of the water and back down. Picture a frog as you do it; this will help you get the form right. Next, to work the upper body, begin by making a scoop with both hands at the surface of the water. Bring your hand scoop below the surface, then scoop up and to one side. Alternate sides to work the obliques on both sides. Start by doing the two moves separately for three minutes each. Once you have mastered the form, do them at the same time to exercise both sets of stomach muscles at once.

As you build strength and endurance, you can add water gloves to increase resistance. You can also do the exercises faster, packing more reps into each three minute period. Form is key, though. Do not sacrifice form for the sake of speed. It is better to do it properly than quickly. Also remember to set fitness goals and work toward them at a gradual pace. Do not push yourself too hard, too fast. You may want to see results fast, but an injury will seriously delay your workout goals. Start small, know your limits, and build gradually.

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