What is the mass of the box?

This problem is similar to the example in Lecture 7 where we used friction to stop motion. The difference is that we need to resolve the applied force in to horizontal and vertical components. Should we use the static or kinetic friction coefficient for this problem?

A. How far up the plane will it go?

B. How much time elapses before it returns to its starting point?

Find the normal force to be able to find the frictional force. We also need to think about the component of the gravitational force down the slope. What we need from this is the acceleration, and as we know the initial velocity $v_{0}$ we can find the distance it travels up the plane using the appropriate equation of motion. For part B we need to again work out the net force (or is it the same as before?) and use a different equation of motion which connects the answer to part B to time.

A. Calculate the coefficient of kinetic friction $\mu$ between block A and the table top.

B. A cat, also of weight $w_A$, falls asleep on top of block A. If block B is now set into downward motion, what is the magnitude of its acceleration?

Part A: Balance the frictional force $\mu w_{A}$ and the gravitational force $w_{B}$

Part B: Find the net force on the object, taking into account your result from part A. Use Newton's second law to find the acceleration, remembering that weight and mass are not the same!

If $m_A$ moves up, and $m_B$ moves down, determine their acceleration.

Find the sum of the forces on each object, taking the direction of the frictional force from the information given in the question about the direction of motion. As in the Atwood machine the tension force should be constant through the rope and the acceleration of both objects should be the same.

A. Assuming a drag force determine the value of the constant b.

B. Assuming a drag force determine the time required for such a drop, starting from rest, to reach 63% of terminal velocity.

Part A: See the notes on Terminal velocity - The velocity where the drag force equals the gravitational force.

Part B: See the notes on the properties of exponential functions.

How many revolutions per minute would a 21 m-diameter Ferris wheel need to make for the passengers to feel “weightless” at the topmost point?

The force analysis is the same as for the looping plane example in the lecture. Once you have $v$ you need to convert it to revolutions per minute, by dividing the length of the circular path by the velocity, and the converting from rev/s to rev/min.

If a curve with a radius of 90 m is properly banked for a car traveling 68 km/h, what must be the coefficient of static friction for a car not to skid when traveling at 96 km/h. (Your numbers may differ).

The angle $\theta$ can be found from the condition required for the banked curve to work without friction ($\tan\theta=\frac{v^2}{rg}$).

Now you need to consider the component of the frictional force which acts in the radial direction $(\mu F_{N}\cos\theta)$. The sum of this and the force provided due to banking must equal the centripetal force at the new higher velocity.

You also need to consider the new equation for the vertical forces.

You need to combine the two equations to eliminate $F_N$ and $m$ (think about how it is done without friction).

Force equations without friction:

$F_{N}\sin\theta-\frac{mv^{2}}{r}$

$F_{N}\cos\theta=mg$

Force equations with friction:

$F_{N}\sin\theta+\mu F_{N}\cos\theta=\frac{mv^{2}}{r}$

$F_{N}\cos\theta-\mu F_{N}\sin\theta=mg$

Divide one equation by the other

$\frac{\sin\theta+\mu\cos\theta}{\cos\theta-\mu\sin\theta}=\frac{v^{2}}{rg}$

How far from the bottom of the chute does the ball land?

A combination of our recent topic, circular motion, with what we did earlier in the course, projectile motion.

If you weigh 685 N on the earth, what would be your weight on the surface of a neutron star that has the same mass as our sun and a diameter of 25 km? (Your numbers may differ).

Find the mass of the person using the acceleration due to gravity at the Earth's surface. Now use Newton's law of gravitation with the mass of the person and sun, and the radius of the Neutron Star.

How far would you have to drill into the Earth, to reach a point where your weight is reduced by 6.5%? Approximate the Earth as a uniform sphere. (Your numbers may differ).

Use the formula from the falling through the Earth example from Lecture 9.