Mechanics

Help Questions

Physics › Mechanics

Questions 1 - 10
1

A toy car is set up on a frictionless track containing a downward sloping ramp and a vertically oriented loop. Assume the ramp is tall. The car starts at the top of the ramp at rest.

What additional piece of information is necessary to calculate the maximum height of the loop if the car is to complete the loop and continue out the other side?

None

The mass of the car

The exact shape of the loop

The value of

The distance between the end of the ramp and entrance to the loop

Explanation

This is an example of conservation of energy. The car starts at the top of the ramp, at height . It has no velocity at this time since it is starting from a rest. Therefore its total energy is where is the mass of the car and is the value of gravitational acceleration.

At the bottom of the loop, all of the potential energy will have been converted into kinetic energy.

As the car traverses the loop and rises above the ground, kinetic energy will be converted back into potential energy. The shape of the loop does not matter in this case -- only the vertical distance between the ground and the car.

In the tallest possible loop, all kinetic energy at the bottom is converted to potential energy at the top. This is the maximum height the car can reach -- there is no additional energy left to continue climbing a taller loop. Therefore, the potential energy at the top of the tallest loop we can build is equal to the kinetic energy at the bottom of the loop. But we have already noted that the kinetic energy at the bottom of the loop is equal to the potential energy at the top of the ramp.

Therefore, we set . We see that and cancel, and we are left with . In other words, the tallest loop you can build is equal to the height of whatever ramp you select. In this example, the tallest loop we can build is . We do not need to know the specific values of or .

2

How far can a person jump while running at and a vertical velocity of ?

Explanation

We know that:

and we are looking for the maximum height (vertical displacement) this person can obtain, so we aren't concerned with .

We can apply the conservation of energy:

Masses cancel, so

Solve for :

(rounded to simplify our calculations)

so let's plug in what we know

. This is our final answer.

3

Fluid flows into a pipe with a diameter of at a rate of . If the other end of the pipe has a cross sectional area of , what is the speed of the fluid as it exits the pipe?

Explanation

This question tests the concept of the continuity equation which stipulates that at steady state the volume flowing into one end of a pipe must be exactly equal to the volume flowing out of the other end. That is, . Note that in this case, we are given the inlet diameter and not cross-sectional area. Thus, we must also find the inlet's cross-sectional area by using the formula for the area of a circle.

4

A baseball has a mass of , but it weighs when completely submerged in water. What is its volume assuming that the density of water is ?

Explanation

We are given the mass of the baseball outside of the water. Using the weight equation with the gravitational constant being and the mass being , the weight of the baseball outside of the water is 4.905 N. (Be careful and convert the mass of the baseball from grams to kilograms since we are using SI units).

The buoyancy force is the difference of the weight of the baseball when it is in the air and when it is in the water. So subtract the two differences:

Now we use the buoyancy equation:

where is the buoyancy force, is the density of water, is the volume of the baseball, and is the gravitational constant. Plug in the known variables and solve for the volume.

and we get

5

Suppose a box is being dragged across the floor due to a rope being pulled on at a angle from the side, as shown in the picture below.

Vt physics friction prob.

If the tension in the rope is and the box accelerates to the right at , what is the coefficient of kinetic friction?

Explanation

To solve this problem, we first need to take into account the forces acting in the vertical direction separately from the forces acting in the horizontal direction.

First, let's start with the vertical direction. Here, the only force force acting downward is the weight of the box. There are two forces acting upwards on the box; one is the normal force and the other is the vertical component of the tension in the rope that is pulling the box. Because the box is not moving in the vertical direction, there is no net force. Thus, the sum of these forces is equal to zero.

Next, let's take a look at the forces acting horizontally on the box. Acting to the right of the box is the horizontal component of the tension in the rope. Acting to the left is the frictional force. Because the box is moving to the right, it must have experienced a net force in this direction. Thus, the sum of these horizontally acting forces will equal a net force.

Now, we can take the expression we obtained for the normal force and substitute it into the expression we obtained for the horizontally acting forces.

Now that we've found an expression for the coefficient of kinetic friction, all we need to do is plug in the values given in the question stem to arrive at the answer.

6

A pendulum is made up of a small mass that hangs on the end of a long string of negligible mass. The pendulum is displaced by and allowed to undergo harmonic motion. What is the angular frequency of the resulting motion?

Explanation

The angular frequency of a simple pendulum is , where is the length of the pendulum.

7

For a simple harmonic motion governed by Hooke's Law, , if is the period then the quantity is equivalent to which of the following?

Explanation

We know that T is the period. The equation for T is for harmonic motion.

Solve for by dividing the equation by on both sides. The result is , which is the answer.

8

Which of these forces is identical to a normal force?

All of these

Force that is perpendicular to the surface

The scale weight of a body

The perpendicular force

The contact force

Explanation

All of these are identical to a normal force.

9

A pendulum is made up of a small mass that hangs on the end of a long string of negligible mass. The pendulum is displaced by and allowed to undergo harmonic motion. What is the angular frequency of the resulting motion?

Explanation

The angular frequency of a simple pendulum is , where is the length of the pendulum.

10

A toy car is set up on a frictionless track containing a downward sloping ramp and a vertically oriented loop. Assume the ramp is tall. The car starts at the top of the ramp at rest.

What additional piece of information is necessary to calculate the maximum height of the loop if the car is to complete the loop and continue out the other side?

None

The mass of the car

The exact shape of the loop

The value of

The distance between the end of the ramp and entrance to the loop

Explanation

This is an example of conservation of energy. The car starts at the top of the ramp, at height . It has no velocity at this time since it is starting from a rest. Therefore its total energy is where is the mass of the car and is the value of gravitational acceleration.

At the bottom of the loop, all of the potential energy will have been converted into kinetic energy.

As the car traverses the loop and rises above the ground, kinetic energy will be converted back into potential energy. The shape of the loop does not matter in this case -- only the vertical distance between the ground and the car.

In the tallest possible loop, all kinetic energy at the bottom is converted to potential energy at the top. This is the maximum height the car can reach -- there is no additional energy left to continue climbing a taller loop. Therefore, the potential energy at the top of the tallest loop we can build is equal to the kinetic energy at the bottom of the loop. But we have already noted that the kinetic energy at the bottom of the loop is equal to the potential energy at the top of the ramp.

Therefore, we set . We see that and cancel, and we are left with . In other words, the tallest loop you can build is equal to the height of whatever ramp you select. In this example, the tallest loop we can build is . We do not need to know the specific values of or .

Page 1 of 18