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Calculation Of Radial And Tangential Force

Learn steps to identify radius of curvature and tangential force formula, practice to calculate the centripetal force by taking components along radial direction equation & centripetal acceleration of the body.

Identification of Radius of Curvature

  • A particle is moving in circular motion, whether in complete circle or in a part of circle, as shown in figure, the radius of the circle is called the radius of curvature.

Steps to Identify Radius of Curvature 

Step 1- Identify the circle and its plane.

For example :  A car moving on a bent road.

Step 2- Locate the center of the circle.

Step 3- The line joining the center and the periphery is known as the radius of curvature.

Illustration Questions

In the given arrangement, what will be the radius curvature after cutting the string at point \(S\)?

A \(PQ\)

B \(PR\)

C \(QR\)

D \(PS\)

×

As the string is cut, it oscillates like simple pendulum, as shown in figure.

image

Identification of circle and its center 

image

Radius of curvature 

\(PQ\) is the radius of curvature.

image

In the given arrangement, what will be the radius curvature after cutting the string at point \(S\)?

image
A

\(PQ\)

.

B

\(PR\)

C

\(QR\)

D

\(PS\)

Option A is Correct

Free Body Diagrams (FBD) for Circular Motion

  • To draw the Free Body Diagrams (FBD) for circular motion,
  1. Represent all forces acting on the body.
  2. In \(FBD\), \(ma_c\) or \(\dfrac{mv^2}{r}\) and \(ma_t\) are not shown because these are not forces.

Illustration Questions

A block is moving in a curved bucket, as shown in figure. Which of the following is the correct \(FBD\) of block?

A

B

C

D

×

\(FBD\) of the block 

image

Option (A) is correct.

A block is moving in a curved bucket, as shown in figure. Which of the following is the correct \(FBD\) of block?

image
A image
B image
C image
D image

Option A is Correct

Calculation of Centripetal Force by taking Components Along Radial Direction

  • In circular motion, centripetal force is the net radial force directed towards center.

For example 

  • Consider a round table which is rotating.
  • A block is placed on the corner of the table.

\(FBD\) of the block

  • Net radial force acting towards the center of table \(=f\)
  • Here, frictional force is holding the block, so that it does not fall from table.
  • Hence, \(f\) is the centripetal force \((F_c)\).
  • Centripetal acceleration is given by

\(a_c=\dfrac{F_c}{m}\)

 

Illustration Questions

The \(FBD\) of a body is shown in figure. Find the centripetal force acting on the body. \(\left[\text{Given:}\;cos\,37°=\dfrac{4}{5},\;cos\,53°=\dfrac{3}{5}\right]\)

A \(10\,N\)

B \(20\,N\)

C \(30\,N\)

D \(40\,N\)

×

\(F_{right}=5×cos\,37°+10×cos\,53°\)

\(F_{right}=5×\dfrac{4}{5}+10×\dfrac{3}{5}\)

\(F_{right}=4+6\)

\(F_{right}=10\,N\)

\(F_{left}=20\,N\)

image

Centripetal force \(=20\,N-10\,N\)

\(=10\,N\)

 

image

The \(FBD\) of a body is shown in figure. Find the centripetal force acting on the body. \(\left[\text{Given:}\;cos\,37°=\dfrac{4}{5},\;cos\,53°=\dfrac{3}{5}\right]\)

image
A

\(10\,N\)

.

B

\(20\,N\)

C

\(30\,N\)

D

\(40\,N\)

Option A is Correct

Calculation of Centripetal Force

  • In circular motion, centripetal force is the net radial force directed towards center.

For example 

  • Consider a round table which is rotating.
  • A block is placed on the corner of he table.

\(FBD\) of the block

  • Net radial force acting towards the center of table \(=f\)
  • Here, frictional force is holding the block, so that it does not fall from table.
  • Hence, \(f\) is the centripetal force \((F_c)\).
  • Centripetal acceleration is given by

\(a_c=\dfrac{F_c}{m}\)

 

Illustration Questions

A ball suspended from ceiling by a light string, is moving along a horizontal circle at an angle \(\theta=37°\), as shown in figure. If tension force acting in the string is \(T=5\,N\), then find the centripetal force. \(\left[\text{Given :}\;sin\,37°=\dfrac{3}{5}\right]\)

A \(3\,N\)

B \(6\,N\)

C \(4\,N\)

D \(10\,N\)

×

\(FBD\) of the ball

image

Centripetal force is the net force acting towards the center of the circular motion.

Thus,

\(F_c=T\,sin\,\theta\)

Given : \(\theta=37°,\;T=5\,N\)

\(F_c=5\,sin\,37°\)

\(=5×\dfrac{3}{5}\)

\(=3\,N\)

A ball suspended from ceiling by a light string, is moving along a horizontal circle at an angle \(\theta=37°\), as shown in figure. If tension force acting in the string is \(T=5\,N\), then find the centripetal force. \(\left[\text{Given :}\;sin\,37°=\dfrac{3}{5}\right]\)

image
A

\(3\,N\)

.

B

\(6\,N\)

C

\(4\,N\)

D

\(10\,N\)

Option A is Correct

Component of Forces along Radial and Tangential Direction in Circular Motion

  • Draw \(FBD\) of the body.
  • Resolve forces along radial and tangential direction of circle.

For example 

  • Consider an arrangement shown in figure, in which a mass \(m\) is attached with two strings.

  • At some instant, when the string \(2\) is cut, the particle starts motion as shown in figure.

  • Identification of the circle and its center 

  • \(FBD\) just after the cutting of string 2 

        Resolve forces along radial direction only.

Illustration Questions

In the arrangement of conical pendulum, which one of the following components of force is along radial direction?

A \(T\,sin\,\theta\)

B \(T\,cos\,\theta\)

C \(mg\,cos\,\theta\)

D \(mg\,sin\,\theta\)

×

Step 1: Identify circle and radius of curvature

image

Step 2: \(FBD\)

image

Step 3: Resolve forces along radial direction 

image

In the arrangement of conical pendulum, which one of the following components of force is along radial direction?

image
A

\(T\,sin\,\theta\)

.

B

\(T\,cos\,\theta\)

C

\(mg\,cos\,\theta\)

D

\(mg\,sin\,\theta\)

Option A is Correct

Direction of Centripetal Force

  • In circular motion, centripetal force is the net radial force directed towards center.

For example 

  • Consider a round table, whose top is rotating.

  • A block is placed on the edge of the table.

\(FBD\) of the block 

  • Net radial force acting towards the center of table \(=f\)
  • Here, frictional force is holding the block, so that it does not fall from table.
  • Hence, \(f\) is the centripetal force \((F_c)\).
  • Centripetal acceleration is given by

\(a_c=\dfrac{F_c}{m}\)

 

Illustration Questions

The \(FBD\) of a body of mass \(m=2\,kg\) is shown in figure. Find the centripetal acceleration of the body, if it is moving in circular motion.

A \(4\,m/s^2\)

B \(3\,m/s^2\)

C \(6\,m/s^2\)

D \(2\,m/s^2\)

×

Net radial force 

\(=10N-4N\)

 \(\text {Centripetal force}=6\,\text{N}\)

\(\text{Centripetal acceleration}=\dfrac{F_\text{centripetal}}{m}\)

\(a_c=\dfrac{6}{2}\)

\(a_c=3\,m/s^2\)

The \(FBD\) of a body of mass \(m=2\,kg\) is shown in figure. Find the centripetal acceleration of the body, if it is moving in circular motion.

image
A

\(4\,m/s^2\)

.

B

\(3\,m/s^2\)

C

\(6\,m/s^2\)

D

\(2\,m/s^2\)

Option B is Correct

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