# Centripetal force problems and solutions pdf

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We know from kinematics that acceleration is a change in velocity, either in its magnitude or in its direction, or both. In uniform circular motion, the direction of the velocity changes constantly, so there is always an associated acceleration, even though the magnitude of the velocity might be constant.

## Centripetal force problem solving

A ball is attached to a string that is 1. It is spun so that it completes two full rotations every second. What is the centripetal acceleration felt by the ball? We are told that it travels in a circle with radius 1. The length of each rotation is just the circumference of the circle:. The ride is a circlular wall that you place your back on. The wall and floor then begin to spin. A student of mass 50kg decides to go on the ride. The coefficient of static friction between the student and wall is 0.

There is a lot going on in this problem and it will take several steps to get to the answer. However, when you boil the question down, we are pretty much asked how fast must the ride spin so that the centripetal force on the student provides enough static friction to keep the student from falling. Let's work through this problem one step at a time. First, let's figure out what minimal static frictional force is required. This force will be equal to the weight of the student; the student's weight will pull downward, while the friction of the wall pushes upward.

Now we can calculate the normal force required to reach that magnitude of frictional force. Note that the vector for the normal force will be perpendicular to the wal, directed toward the center of the circlel. This normal force is the minimum centripital force required to keep the student pinned to the wall.

We can convert this to centripital acceleration:. This is the velocity that the outer wall of the ride must be spinning at.

Since we know the radius of the ride, we can convert this velocity into a maximum period, the final answer:. If you weren't sure how to come about this equation, just think about your units. Therefore, you need to cancel out the meter and get the second on top. Candy companies have long strived to catch the attention of children. One item that does this particularly well is the gumball machine. Substituting in our expressions for each variable and removing initial kinetic energy and final potential energy which will each be zero , we get:.

The only variable we don't have at this point is the distance the gumball travels. However, we can calculate it knowing the height of the track and its slope. We can imagine that the spiral track is unwound, creating a right triangle with an angle of 10 degrees and a height of 1. For this triangle, the hypotenuse will be the total distance of the track. A boy is riding a merry-go-round with a radius of. What is the centripetal force on the boy if his velocity is? We are given the mass, radius or rotation, and the linear velocity.

Using these values, we can find the centripetal force. There are two total forces in the system: gravity and tension. It is important to note that the tension isn't only resulting from gravity; it also includes the centripetal force required to keep the ball in circular motion.

Thinking practically, we can say that the greatest tension will be when the ball is at its lowest point gravity and tension are in opposite directions. The formula for that is:. We know the radius length of the string , so we need to develop an expression for velocity. The centripetal force acting on the car is. If the car's velocity is doubled, what is the new centripetal force required for the car to drive on the circular track?

Centripetal force is proportional to the square of velocity. If friction between the surfaces is negligible, what is the minimum height that the block can start off at so that it will go all the way around the loop without falling off? To start off with, it is useful to consider the energy of the system. Initially, the block is at a certain unknown height, and will thus have gravitational potential energy. Since we are assuming that there is no friction in this case, then we know that total mechanical energy will be conserved.

Therefore, all of the gravitational potential energy contained in the block will become kinetic energy when it slides down to the bottom of the loop. But, once the block begins to slide up the loop, it will lose kinetic energy and will regain some gravitational potential energy.

Therefore, at the top of the loop, the block will have a combination of kinetic and gravitational potential energy whose sum is equal to the initial energy of the system.

In addition to considering energy, it is also necessary to consider the forces acting on the block in this scenario. When at the top of the loop, the block will experience a downward force due to its weight, and another downward force due to the normal force of the loop on the block.

Furthermore, because the block is traveling along a circular path while in the loop, it will experience a centripetal force. At the top of the loop, the centripetal force will be due to a combination of the weight of the block as well as the normal force.

We're looking for a starting height that will just allow the block to travel around the loop. The minimum height will be the height such that the block will just start to fall off.

When falling off, the block will no longer be touching the loop and therefore, the normal force will be equal to zero. This is the situation we are looking for, and since the normal force is zero, only the block's weight will contribute to the centripetal force at the top of the loop.

We can plug this expression into the previous energy equation. A person is crossing a canyon by swinging on a vine, as shown in the given figure. At that instant, what is the tension force in the vine? Instead, they add to a net force pointing towards the center of the circle that the person is making, which is up at the place where the vine is attached to the tree.

Draw a vector diagram:. Then write the equation about the lengths of the vectors: the length of the gravity vector plus the length of the net force vector equals the length of the tension vector:.

The net force for an object undergoing circular motion is mass times speed squared divided by the radius of the circle. Plug in the numbers, and solve for the tension:. We can determine the centripetal force exerted by the nucleus on an electron. The mass of an electron is roughly. Since we are given the diameter of an atom, its radius will be. Imagine a car driving over a hill at a constant speed. Once the car has reached the apex of the hill, what is the direction of the acceleration?

If we imagine the hill as a semi-circle, it appears that the car is moving along a circle. At the apex of the hill, the car's acceleration points downwards as this points towards the center of the circle. If an object travels in a circular fashion, at a constant speed, the direction of acceleration is always towards the center of the circle.

This type of acceleration arises do to the change in velocity. Although the speed is constant, the direction changes. If you've found an issue with this question, please let us know. With the help of the community we can continue to improve our educational resources.

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Hanley Rd, Suite St. Louis, MO Subject optional. Home Embed. Email address: Your name:. Possible Answers:. Correct answer:. The length of each rotation is just the circumference of the circle: The velocity can be found by multiplying that distance by the frequency: Now we have all of our variables and can plug into our first equation:.

Report an Error. Explanation : There is a lot going on in this problem and it will take several steps to get to the answer. We can convert this to centripital acceleration: We can now convert centripital acceleration to a translational velocity using the equation: Rearranging for velocity, we get: This is the velocity that the outer wall of the ride must be spinning at.

Since we know the radius of the ride, we can convert this velocity into a maximum period, the final answer: If you weren't sure how to come about this equation, just think about your units. Explanation : We can use the expression for conservation of energy to solve this problem: Substituting in our expressions for each variable and removing initial kinetic energy and final potential energy which will each be zero , we get: Rearranging for final velocity: The only variable we don't have at this point is the distance the gumball travels.

Now that we have all of our variables, we can solve for the final velocity: We can then use this to calculate the centripetal force on the gumball:. Explanation : For this problem, we use the centripetal force equation: We are given the mass, radius or rotation, and the linear velocity. The formula for that is: We know the radius length of the string , so we need to develop an expression for velocity. Substituting this into the expression for centripetal acceleration: Substituting this back into the equation for tension, we get: We have all of these values, allowing us to solve:.

Explanation : The equation for centripetal force is:. Explanation : To start off with, it is useful to consider the energy of the system. Draw a vector diagram: Then write the equation about the lengths of the vectors: the length of the gravity vector plus the length of the net force vector equals the length of the tension vector: The net force for an object undergoing circular motion is mass times speed squared divided by the radius of the circle.

## AP Physics 1 : Centripetal Force and Acceleration

Example — The vertical section of a road over a bridge in the direction of its length is in the form of an arc of a circle of radius Find the greatest velocity at which a car can cross the bridge without losing contact with the road at the highest point if the c. A flyover bridge is in the form of a circular arc of radius 30 m. Find the limiting speed at which a car can cross the bridge without losing contact with the road at the highest point.

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Data from high-technology firms in Hong Kong were used to investigate whether the outcomes of problem-solving processes solutions found, problem-solving speed, and solution quality mediated the effects of centrifugal forces decentralization, free flow of information, and reach and centripetal forces connectedness, temporal pacing, project leader expertise, and superordinate goal on product development performance development speed and product quality. Centrifugal and centripetal forces were indirectly related to performance through problem-solving outcomes, although some direct effects of these forces were also found. The results suggest a more complex model of product development than previously envisaged.

If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Donate Login Sign up Search for courses, skills, and videos. Science Physics library Centripetal force and gravitation Centripetal forces. Centripetal force problem solving.

### Centripetal force – problems and solutions

A gram ball, attached to the end of a cord, is revolved in a horizontal circle with an angular speed of 5 rad s K nown :. Wanted : The centripetal force. Solution :.

In Motion in Two and Three Dimensions , we examined the basic concepts of circular motion. An object undergoing circular motion, like one of the race cars shown at the beginning of this chapter, must be accelerating because it is changing the direction of its velocity. We proved that this centrally directed acceleration, called centripetal acceleration , is given by the formula.

Problem : A 2 kg ball on a string is rotated about a circle of radius 10 m. The maximum tension allowed in the string is 50 N. What is the maximum speed of the ball? The centripetal force in this case is provided entirely by the tension in the string. If the maximum value of the tension is 50 N, and the radius is set at 10 m we only need to plug these two values into the equation for centripetal force:. Problem : During the course of a turn, an automobile doubles its speed.

When the magnitude of the centripetal force is 20 N, the speed with which the child can swing the money is at a maximum. For circular motion, the centripetal.