Spring Force - AP Physics 1

Since the block is motionless, we know that our forces will cancel out:

There are three forces in play: one from each spring, as well as the force of gravity. If we assume that forces pointing up are positive, we can write:

Plugging in expressions for each spring force, we get:

Rearring for the displacement of the top spring, we get:

Since the block is motionless, we can assume that the forces cancel out:

If we designate any forces pointing downward as positive, we can write:

Inserting expressions for each force, we get:

Rearranging for mass, we get:

There are two ways to solve this problem: the first involves calculating the spring constant and the second does not. We'll go through both methods.

Calculating Spring Constant

We can use the expression for the force of a spring:

At equilibrium, the force of the spring equals the force of gravity:

Rearranging for the spring constant and plugging in values, we get:

Now, apply this equation when the spring is on a different planet:

Rearranging for displacement and plugging in values, we get:

Without Calculating Spring Constant

We can write the force equation for each scenario. Let the subscript 1 denote Earth, and 2 denote the other planet:

Using these equations, we can set up a proportion:

Rearranging for the displacement of scenario 2 and plugging in values:

Since the displacement of the spring is at equilibrium, we can write:

There are three forces we can account for: spring force, gravitational force, and the additional force resulting from the acceleration of the elevator. If we assume that forces pointing upward are positive, we can write:

If you are unsure whether the force resulting from the acceleration of the elevator will be positive or negative, think about the situtation from personal experience: When an elevator begins to accelerate upward, your body feels heavier. Thus, the force adds to the normal gravitational force.

Substituting expressions for each force, we get:

Rearrange to solve for displacement:

The maxmimum force applied by the spring will occur when the mass is at its maximum displacement. Since we know the energy of the system, we can calculate displacement using the following expression:

Rearranging for displacement, we get:

We can use this, along with the expression for force applied by a spring:

Substitute our first expression into our second and simplify:

We have values for each variable, allowing us to solve for the force:

Note that the mass of the block is irrelevant to the problem. Mass does not effect displacement or force applied by the spring; it only affects the velocity of the block at different points.

We simply need to alter the expression for the force of the springs to solve this problem. The following is the original expression:

Since each spring has the same costant, they actually act as one large spring with the same, original costant. Therefore the value of in this equation is the total displacement. Multiplying the displacement of each single spring by the number of total springs will give us this total displacement:

Here, is the number of springs, and this new displacement, , is the displacement of each spring. Rearranging for the number of springs, we get:

We can use force equilibrium to begin our derivation:

There are two vertical forces in play: gravity and total spring force. Since net force is zero, we know that these two general forces are equal to each other:

Substituting in expressions for each force, we get:

There are two things to note about the total spring force. The first is that we multiply it by 40 because there are 40 individual springs. Second, we multiply the force by the sine of the angle, because we only want to know the vertical force applied by the springs.

Rearranging for the displacement of the springs, we get:

We have values for each variable, allowing us to solve:

The base equation for position when undergoing simple harmonic motion is:

First, solve for the phase constant.

Plug all the variables into the equation and solve.

Recall Hooke's Law, which states:

Here, is the force exerted by the spring, is the spring constant, and is the displacement from the spring's rest position. This equation tells us that the force exerted is directly proportional to the displacement. We don't need to solve for to determine the magnitude of the force on the spring stretched 10m. We can instead come up with a proportionality such that:

Here, and are forces applied on the string and and are the displacements of the spring from its respect position respectively. We assume that a stretched spring will have a positive displacement, whereas a shrunken spring will have a negative displacement. However, since we're looking for the magnitude of the force, regardless of direction, the direction of the displacement doesn't matter. Therefore, we can write the proportion as:

In our case:

Hooke's law states that the spring force is equal to the product of the spring constant and the displacement of the spring:

The force is negative because it acts in the direction opposite of the displacement from the equilibrium position (i.e. when we stretch we do so in the positive direction). We are given the force and the displacement, so we just solve for k:

This question is giving us the amount of force needed to displace an object attached to a spring, and is asking us to calculate the spring constant. Thus, we will need to make use of Hooke's law.

The above equation, Hooke's law, tells us that the restoring force of the spring is related to the displacement of the attached object. It's important to note that in the question stem, we are told that an external force of is needed to displace this object. Thus, the restoring force of the spring, due to Newton's third law, has the same magniture of the applied force but in the opposite direction. Thus, the restoring force of the spring is equal to . Plugging this value into Hooke's law, as well as the displacement of , yields:

The spring is held outside of equilibrium at rest by a certain force. That means that this force is equal in magnitude to the restoring force of the spring. By Hooke's law, the magnitude of the restoring force is given by

Where is the spring constant, and is the displacement of the spring. If is the force that keeps the spring stretched, then . Solving for gives us

Now we need to find the work required to stretch the spring further from to . Well, we know that the work done is equal to the negative of the change in potential energy of the spring by the work-energy theorem. The change in energy is given by:

Therefore the work done is just the negative of the above, or

Where is the spring constant

is the compression of the spring

is the mass of the object.

is the gravity constant, which will be treated as a negative number.

Solve for :

Plug in values:

Where and are the respective spring constants, is the stretch length, and is the gravity constant, which is a negative as the vector is pointing down.

Since

Substitute and plug in values:

Solving for :

Determine the net forces on the submarine:

Plug in values:

Determine what forces are acting on the submarine:

Plug in values

*Note: Acceleration due to gravity is going down so it is a negative

Solve for :

Each spring will be subject to the same force, and since they have the same spring constant, stretch the same amount. Thus:

Total stretch:

Stretch of one spring:

Use Hooke's law:

The force will be equal to the force of gravity on the mass:

Solve for :

Use Hooke's law:

and

Wide spring:

Narrow spring:

From given information:

Substitute:

Summation of forces:

Where is pointing down and thus a negative value. Convert to and plug in values:

Solve for :

Plug back into the following formula and solve for the spring constant of the wider spring.

The force of the spring in relationship to strain is independent of direction. Thus, the same force pulling on the spring would result in an equal amount of length change, albeit by stretching instead of compressing.

Use Hooke's law:

and

Wide spring:

Narrow spring:

From given information:

Substitute:

Summation of forces:

Where is pointing down and thus a negative value. Convert to and plug in values:

Solve for :