Two blocks are connected by a rope which runs over a frictionless pulley of negligible mass. One block hangs from the rope, while the other rests on a frictionless plane inclined at an angle of $\theta$ to the horizontal.

A. Find an expression for the acceleration, $a$, of $m_{1}$ as a function of $m_{1}$, $m_{2}$, $g$ and $\theta$. Specify which direction is positive (i.e up or down).

Taking up positive.

$m_{1}a=T-m_{1}g$

$m_{2}a=m_{2}g \sin\theta-T$

$(m_{1}+m_{2})a=m_{2}g \sin\theta-m_{1}g$

$a=\frac{m_{2}\sin\theta-m_{1}}{m_{1}+m_{2}}g$

For parts B-D consider a case where $m_{1}=1\mathrm{kg}$ and $m_{2}=3\mathrm{kg}$.

B.For which value of the angle $\theta$ would the system remain at rest after it has been released?

This is when the numerator of the expression found in B is zero.

$3\sin\theta=1$

$\sin\theta=\frac{1}{3}$

$\theta=19.47^{o}$

C. If $\theta=30^{o}$ what is the velocity and displacement of $m_{1}$, 0.5 s after the system has been released from rest? Specify which direction is positive (i.e up or down).

$a=\frac{\frac{3}{2}-1}{4}g=\frac{g}{8}=1.225\mathrm{m\,s^{-2}}$

$v=1.225\times0.5=0.6125\mathrm{m\,s^{-1}}$

$x=0.5\times1.225\times0.5^{2}=0.15\mathrm{m}$

D.

What is the magnitude of the tension in the rope during the motion in part (C)?

$T=(1+\frac{1}{8}g)=11.025\mathrm{N}$

A. Which satellite is faster?

B. By what factor?

Make the equality between centripetal and gravitational force. Don't forget that altitude and distance from the center of the Earth are not quite the same thing!

For GPS satellites:

A. Determine the speed of each satellite.

B. Determine the period of each satellite.

We will do a problem like this in class.

A frictionless incline. You are asked to find the work done by various forces during the motion of the block up the incline.

For each force you need to find the component of the force in the direction of motion. This times the displacement gives you the work done by that force.

To find the velocity, find the net work done on the body and use the work energy theorem.

Find the scalar product of the two vectors.

You need to either deduce the angle between the vectors and use $\vec{V_{1}}\cdot\vec{V_{2}}=V_{1}V_{2}\cos\theta$

or you can find the component of $\vec{V_{2}}$ in the z direction and multiply that by $V_{1}$ which is entirely in the z direction.

What is the work done when x goes from 0.1 to $\infty$?

As this force depends on distance you will need to integrate the function to get the work done:

$W=\int_{0.1}^{\infty}Ae^{-kx}dx$

Part A. Given mass calculate the work done against gravity.

Use conservation of energy to solve this!

Part B. What force needs to be exerted on the pedals given the distance the bike moves during one turn of the pedals? \ The work done on the pedals must be the same as the work done against gravity. As you can work out what the work done against gravity is in one cycle and you know how far the feet move in one cycle you can get the force on the feet. The power of gears!

An inclined plane with friction. The total mechanical energy is **not** conserved, but we can quantify the work done by friction which is lost to the motion.

The equation

$\frac{1}{2}mv_{1}^2+mgh_{1}=\frac{1}{2}mv_{2}^2+mgh_{2}+F_{fr}d$

can be applied to both parts of the motion

This problem combines ideas about conservation of energy and centripetal acceleration.

Part A. Determine the minimum release height.

The minimum required velocity is when the centripetal acceleration at the top of the loop is equal to the gravitational force. (Examine the vertical circle or looping plane diagram to see why.) Then use conservation of energy to relate the mechanical energy at the top of the loop to the original potential energy, mgh. Don't forget that the potential energy is not zero at the top of the loop!

Part B,C,D. The condition at any point in the circle is that the net force in the radial direction must be equal to the centripetal force. Conservation of energy will tell you what the velocity is, and hence what the centripetal force is. You'll also need to take into account the direction of other forces on the object. Also you'll need to you results from part A to remove g from the problem.

For A refer to the equation for gravitational potential energy at large distances.

For B get the velocity from the equation for the velocity of a circular orbit.

For C the work done should be equal to the **total** change in energy from orbit to another.

Part A

Use conservation of energy, kinetic energy at bottom is the converted to potential energy at top.

Part B

Find the amount of energy release in one second. (You had to calculate the energy of one rock in Part A.)