### All AP Physics 1 Resources

## Example Questions

### Example Question #1 : Spring Force

A horizontal spring with a spring constant of is attached to a frictionless surface. A block of mass 2kg is attached to the end of the spring. A man spends 20J of energy to compress the spring. How far from equilibrium is the block?

**Possible Answers:**

**Correct answer:**

You simply need to know the formula for the potential energy stored in a spring to solve this problem. The formula is:

where k is the spring constant, and x is the distance from equilibrium

We can rearrange this to get:

Plugging in our values, we get:

### Example Question #1 : Spring Force

Consider the following system:

Both springs have a constant of and the block is motionless. If the bottom spring is compressed past its equilibrium and the block has a mass of , how far is the top spring stretched past its equilibrium?

**Possible Answers:**

**Correct answer:**

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:

### Example Question #1 : Spring Force

Consider the following system:

Both springs have a constant of and the block is motionless. If the top spring is stretched past its point of equilibrium and the bottom spring is compressed past its point of equilibrium, what is the mass of the block?

**Possible Answers:**

**Correct answer:**

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:

### Example Question #2 : Spring Force

A spring attached to the ceiling has a block of mass attached to the other end. On Earth, the displacement of the spring when at equilibrium is . What is the displacement of the spring at equilibirum when this system is on a planet that has a gravitational acceleration of

**Possible Answers:**

**Correct answer:**

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:

### Example Question #1 : Spring Force

A spring hanging from the ceiling of an elevator has a spring constant of and a block attached to the other end with a mass of . If the elevator is accelerating upward at a rate of and the spring is in equilibirum, what is the displacement of the spring?

**Possible Answers:**

**Correct answer:**

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:

### Example Question #2 : Spring Force

A horizontal spring with a constant of is attached to a wall at one end and has a block of mass attached to the other end. If the system has of potential energy and the block is on a frictionless surface, what is the maxmimum force applied by the spring?

**Possible Answers:**

**Correct answer:**

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.

### Example Question #1 : Spring Force

A series of horizontal springs are attached end to end. The far left spring is attached to a wall. The constant of each spring is . If a tensile force of applied to the right end of the series of springs results in a displacement of for each spring, how many springs are in the series?

**Possible Answers:**

**Correct answer:**

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:

### Example Question #8 : Spring Force

A circular trampoline has springs around the outer edge, each with a constant of . If a child of mass depresses the tramopline such that each spring is at an angle of below horizontal, what is the displacement of each spring?

**Possible Answers:**

**Correct answer:**

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:

### Example Question #1 : Spring Force

A block is attached to a spring with spring constant . The block is pulled away from the equilibrium and released. Where is the block 3 seconds after this occurs? (You may treat the equilibrium as the zero position and a stretched spring as a positive displacement)

**Possible Answers:**

**Correct answer:**

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.

### Example Question #1 : Spring Force

Find the magnitude of the force exerted by a spring on an object that's 10m extended from the rest position, if it exerts 20N of force on the same object that has shrunk 5m from its original position.

**Possible Answers:**

**Correct answer:**

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:

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