Centripetal force – problems and solutions

1. A 200-gram ball, attached to the end of a cord, is revolved in a horizontal circle with an angular speed of 5 rad s^{-1}. If cord’s length is 60 cm, what is the centripetal force?

__K____nown :__

Object’s mass (m) = 200 gr = 200/1000 kg = 2/10 kg = 0.2 kg

Angular speed (ω) = 5 rad/s

Cord’s length = radius (r) = 60 cm = 60/100 m = 0.6 m

__Wanted :__ The centripetal force

__Solution :__

The centripetal force is the resultant force that causes the centripetal acceleration.

The equation of the centripetal force :

∑F = m a_{s}

∑F = m v^{2}/r = m ω^{2} r

∑*F = **Centripetal force**, m = **object’s mass**, v = **linear velocity**, **ω = **angular velocity**, **r = **radius.*

∑F = m ω^{2} r = (0.2)(5)^{2 }(0.6) = (0.2)(25)(0.6) = 3 N

2. A stone attached at the end of a cord and rotated in a horizontal circle by a student. If the final speed of the stone = 2 x the initial speed, then what is the centripetal force.

__Known :__

Stone’s mass = m

Stone’s speed = v

Cord’s length = radius = r

__Wanted:__ The centripetal force

__Solution :__

3. A curve road of radius R is designed so that a car traveling at speed 10 ms^{–1 }can negotiate the turn safely. The coefficient of static friction between car and road = 0.5. What is the radius? Acceleration due to gravity (g) = 10 ms^{–2}.

__Known :__

Speed (v) = 10 m/s

The coefficient of static friction between car and road (μ_{s}) = 0.5

Acceleration due to gravity (g) = 10 m/s^{2}

__Wanted:__ Radius

__Solution :__

The only one force in the horizontal direction is the force of static friction. The equation of the static friction :

4. The coefficient of static friction between tire and road is 0.4. If acceleration due to gravity is 10 m/s^{2}, what is the maximum speed so the car can turn without skidding out of a curved path.

__Known :__

Coefficient of static friction (μ_{s}) = 0.4

Acceleration due to gravity (g) = 10 m/s^{2}

Radius of path (R) = 40 meters

__Wanted:__ maximum speed (v)

__Solution :__

**The equation of Newton’s second law in uniform circular motion :**

*ΣF = **centripetal force** = **net force**, m = m**ass**, a*_{s }*= **centripetal acceleration**, v = **linear speed**, R = **radius of path*

**Centripetal force**

Centripetal force is the net force which produces centripetal accelerations. In this case, the centripetal force is the force of static friction.

The equation of the force of static friction :

*μ*_{s}* = **coefficient of static friction**, w = **weight**, m = mass, g = **acceleration due to gravity*

**The maximum speed ****(v) :**

**What is centripetal force?****Answer:**Centripetal force is the force that acts on an object moving in a circular path, directed towards the center of the circle or path. It is responsible for keeping the object in its circular motion.

**How does centripetal force differ from centrifugal force?****Answer:**Centripetal force is a real force that acts on an object to keep it moving in a circular path. Centrifugal force, on the other hand, is a perceived force that appears to push an object away from the center of a circle when observed from a rotating frame of reference. Centrifugal force is often termed a “fictitious” or “pseudo” force.

**What happens if centripetal force on a rotating object is suddenly removed?****Answer:**If the centripetal force on a rotating object is suddenly removed, the object will move in a straight line tangent to the circular path at the point where the force was removed, following Newton’s first law of motion.

**How does the required centripetal force change with the radius of the circular path for a constant speed?****Answer:**The required centripetal force is inversely proportional to the radius of the circular path. If the speed remains constant but the radius increases, the centripetal force needed decreases, and vice versa.

**How does the centripetal force change if the speed of an object in circular motion is doubled?****Answer:**The centripetal force is directly proportional to the square of the speed of the object. So, if the speed is doubled, the centripetal force required becomes four times greater.

**Can gravity act as a centripetal force?****Answer:**Yes, gravity can act as a centripetal force. A prime example is the gravitational force between the Earth and the Moon, which keeps the Moon in its orbit around the Earth.

**Why do you feel pushed to the side when taking a sharp turn in a car?****Answer:**When a car takes a sharp turn, the centripetal force acts towards the center of the turn, keeping the car on its circular path. However, your body tends to continue in a straight line due to inertia. This outward push you feel is a reaction to the centripetal force and is often mistaken as “centrifugal force.”

**How does the mass of an object affect the centripetal force required for circular motion at a constant speed?****Answer:**The required centripetal force is directly proportional to the mass of the object. If the mass of the object is doubled, the centripetal force needed also doubles, given that the speed and radius remain constant.

**What role does friction play in the centripetal force when a car turns?****Answer:**When a car turns, it is the frictional force between the tires and the road that provides the necessary centripetal force to keep the car in a curved path. If there were no friction (e.g., on an icy road), the car would be unable to make the turn and would slide outwards.

**Can tension in a string act as a centripetal force?**

**Answer:**Yes, tension in a string or rope can act as a centripetal force. For example, when an object is twirled around in a circular path using a string, the tension in the string provides the necessary centripetal force to keep the object moving in the circle.