Informative line

### 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.

#### 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.

Identification of circle and its center

$$PQ$$ is the radius of curvature.

### In the given arrangement, what will be the radius curvature after cutting the string at point $$S$$?

A

$$PQ$$

.

B

$$PR$$

C

$$QR$$

D

$$PS$$

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.

#### 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

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?

A
B
C
D

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

$$a_c=\dfrac{F_c}{m}$$

#### 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 $$40\,N$$

×

$$F_{right}=5×cos\,37°+10×cos\,53°$$

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

$$F_{right}=4+6$$

$$F_{right}=10\,N$$

$$F_{left}=20\,N$$

Centripetal force $$=20\,N-10\,N$$

$$=10\,N$$

### 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

$$40\,N$$

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

$$a_c=\dfrac{F_c}{m}$$

#### 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

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. $$\left[\text{Given :}\;sin\,37°=\dfrac{3}{5}\right]$$

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.

#### 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

Step 2: $$FBD$$

Step 3: Resolve forces along radial direction

### 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

$$mg\,sin\,\theta$$

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

$$a_c=\dfrac{F_c}{m}$$

#### 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 $$2\,m/s^2$$

×

$$=10N-4N$$

$$\text {Centripetal force}=6\,\text{N}$$

$$\text{Centripetal acceleration}=\dfrac{F_\text{centripetal}}{m}$$

$$a_c=\dfrac{6}{2}$$

$$a_c=3\,m/s^2$$

### 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.

A

$$4\,m/s^2$$

.

B

$$3\,m/s^2$$

C

$$6\,m/s^2$$

D

$$2\,m/s^2$$

Option B is Correct