Examples

The following examples are typical physics problems not unlike what you might encounter on a quiz or hour exam.

1. Classsic inclined plane
   

   
   

   $$\sum F_x = mg\sin(\theta) = m a_x$$ (2.14)

   $$\sum F_y = N - mg\cos(\theta) = m a_y = 0$$ (2.15)

   
   

   or

   
   

   $$a_x = g\sin(\theta)$$ (2.16)

   $$a_y = 0$$ (2.17)

   
   

   With this in hand, we can now evaluate the trajectory of the particle (using the solutions developed for one dimensional motion) sliding on the incline from a knowledge of its initial conditions.

2. Classic pulley problem (or Atwood's machine).
   

   We imagine the motion to be ``one-dimensional'' along a line wrapped around the massless pulley. (Don't worry, we do the same problem with a massive pulley later.) We then get (for each mass):

   
   

   $$\sum F_1 = m_1g - T = m_1 a$$ (2.18)

   $$\sum F_2 = T - m_2g = m_2 a.$$ (2.19)

   
   

   If we solve these equations simultaneously, (by adding them, for example) we get:

   
   

   $$m_1g - m_2g = m_1a + m_2a$$ (2.20)

   $$a = \frac{m_1 - m_2}{m_1 + m_2}g$$ (2.21)

   
   

   Again, we can now evaluate the trajectory of the pair of masses (using the solutions developed for one dimensional motion) rolling around the pulley from a knowledge of the initial conditions.

3. Inclined plane with pulley
   

   The pulley is massless, the plane frictionless. Combining the methods of the two problems, you should be able to show that:

   
   

   $$m_2 g - m_1\sin(\theta)g = m_1a + m_2a$$ (2.22)

   
   
   so that
   

   $$a = \frac{m_2 - m_1 \sin(\theta)}{m_1 + m_2}g$$ (2.23)

   
   

   From a knowledge of $a$, you should be able to find $T$ in the string, too.

4. Spring and mass in equilibrium
   

   We will learn how to really solve this problem later. Right now we will just write down Newton's laws for this problem so we can find $a$. Let the $x$ direction be up. Then:

   
   

   $$\sum F_x = -k(x - x_0) - mg = m a_x$$ (2.24)

   
   

   or (with $\Delta x = x - x_0$, so that $\Delta x$ is negative as shown)

   
   

   $$a_x = -\frac{k}{m}\Delta x - g$$ (2.25)

   
   

5. Tether Ball at angle $\theta$, string of length $L$. Find $T$ and $v$ of ball.
   

   We note that if the ball is moving in a circle of radius $r = L \sin\theta$, its centripetal acceleration must be $a_r = -\frac{v^2}{r}$. Since the ball is not moving up and down, the vertical forces must cancel. The only ``real'' forces acting on the ball are gravity and the tension $T$ in the string. Thus in the y-direction we have:
   

   $$\sum F_y = T\cos\theta - mg = 0$$ (2.26)

   
   
   and in the x-direction (the minus r-direction, as drawn) we have:
   

   $$\sum F_r = T\sin\theta = ma_r = \frac{mv^2}{r}.$$ (2.27)

   
   

   Thus
   

   $$T = \frac{mg}{\cos\theta},$$ (2.28)

   
   
   
   

   $$v^2 = \frac{Tr\sin\theta}{m}$$ (2.29)

   
   
   or
   

   $$v = \sqrt{gL \sin\theta\tan\theta}$$ (2.30)

   
   

6. Weight in an elevator
   

   Apparent weight in an elevator is pretty easy. The evevator accelerates up (or down) at some net rate $a_y$. Now, if you are riding in the elevator, you must be accelerating up with the same acceleration. Therefore the net force on you must be
   

   $$\sum F_y = ma_y$$ (2.31)

   
   
   That net force is made up of two ``real'' forces: The force of gravity which pulls you down, and the (normal) force exerted by all the molecules in the scale on the soles of your feet. This latter force is what the scale (which reads the net force applied to/by its top surface, remember Newton's third law) indicates as your "weight".

   Thus:
   

   $$\sum F_y = ma_y = N - mg$$ (2.32)

   
   
   or
   

   $$N = w = mg + ma_y$$ (2.33)