Practice problems and tests cannot possibly cover every situation that you may be asked to understand in physics. However, some categories of topics come up again and again, so much so that they might be worth some extra review. And that's exactly what these lessons are for - to give you a focused, intensive review of a few of the most essential physics topics.

Extra drill on difficult but frequently tested topics are:

- Tension Extra Drill Problems for AP Physics B & C
- Electric and Magnetic Fields Extra Drill Problems for AP Physics B & C
- Inclined Planes Extra Drill Problems for AP Physics B & C
- Motion Graphs Extra Drill Problems for AP Physics B & C
- Simple Circuits Extra Drill Problems for AP Physics B & C

We call them "drills" for a reason. They are designed to be skill-building exercises and as such, they stress repetition and technique. Working through these exercises might remind you of playing scales if you're a musician or of running laps around the field if you're an athlete. Not much fun, maybe a little tedious, but very helpufl in the long run.

The questions in each drill are all solved essentially the same way. Don't just do one problem after the other.....rather, do a couple, check to see that your answers are right, and then half an hour or a few days later, do a few more, just to remind yourself of the techniques involved.

Below are inclined plances problems.

### How to Do It

Use the following steps to solve these kinds of problems: 1) Draw a free-body diagram for the object (the normal force is perpendicular to the plane; the friction force acts along the plane, opposite the velocity); 2) break vectors into components, where the parallel component of weight is *mg*(sin *θ*); 3) write Newton's second law for parallel and perpendicular components; and 4) solve the equations for whatever the problem asks for.

Don't forget, the normal force is NOT equal to *mg* when a block is on an incline!

### The Drill

Directions: For each of the following situations, determine:

- the acceleration of the block down the plane
- the time for the block to slide to the bottom of the plane

In each case, assume a frictionless plane unless otherwise stated; assume the block is released from rest unless otherwise stated.

### The Answers (A Step-by-Step Solution to #1 is below.)

*a*= 6.3 m/s^{2}, down the plane.*a*= 4.9 m/s^{2}, down the plane.*a*= 5.2 m/s^{2}, down the plane.*a*= 4.4 m/s^{2}, down the plane.- Here the angle of the plane is 27° by trigonometry, and the distance along the plane is 22 m.
*a*= 6.3 m/s^{2}, down the plane.*a*= 6.3 m/s^{2}, down the plane.- This one is complicated. Since the direction of the friction force changes depending on whether the block is sliding up or down the plane, the block's acceleration is NOT constant throughout the whole problem. So, unlike problem #7, this one can't be solved in a single step. Instead, in order to use kinematics equations, you must break this problem up into two parts: up the plane and down the plane. During each of these individual parts, the acceleration is constant, so the kinematics equations are valid.
- up the plane:
- down the plane:

*a*= 6.8 m/s^{2}, down the plane.*t*= 0.4 s before the block turns around to come down the plane.*a*= 1.5 m/s^{2}, down the plane.*t*= 5.2 s to reach bottom.So, a total of

*t*= 5.6 s for the block to go up and back down.

*t* = 2.5 s

*t* = 2.9 s

*t* = 2.8 s

*t* = 3.0 s

*a* = 4.4 m/s^{2}, down the plane.

*t* = 3.2 s

*t* = 1.8 s

*t* = 3.5 s

### Step-by-Step Solution to #1:

*Step 1:* Free-body diagram:

*Step 2:* Break vectors into components. Because we have an incline, we use inclined axes, one parallel and one perpendicular to the incline:

*Step 3:* Write Newton's second law for each axis. The acceleration is entirely directed parallel to the plane, so perpendicular acceleration can be written as zero:

*mg*sin

*θ*– 0 =

*ma*.

*F*

_{N}–

*mg*cos

*θ*= 0.

*Step 4:* Solve algebraically for a. This can be done without reference to the second equation. (In problems with friction, use *F _{f}* = μ

*F*

_{N}to relate the two equations.)

*a*=

*g*sin

*θ*= 6.3 m/s

^{2}.

To find the time, plug into a kinematics chart:

*v*= 0

_{o}*v*= unknown

_{f}- Δ

*x*= 20

*m*

*a*= 6.3 m/s

^{2}

*t*= ???

Solve for *t* using the second star equation for kinematics (**): Δ*x* = *v _{o}t* + 1/2

*at*

^{2}, where

*v*is zero;

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