Torque MCQ Quiz in தமிழ் - Objective Question with Answer for Torque - இலவச PDF ஐப் பதிவிறக்கவும்

CONCEPT:

• Rotational motion: When a block is moving about a fixed axis on a circular path then this type of motion is called rotational motion.
• Torque (τ): It is the twisting force that tends to cause rotation.
  • The point where the object rotates is known as the axis of rotation. 
• Mathematically it is written as,

⇒ τ = r F sin θ 

Where r is the distance between the axis of rotation and point of application of force, F is force and θ is the angle between r and F.

EXPLANATION:​

Given: Length of the rod = L, the mass of the rod = M, force on one end = F.

• We know that torque, 

​\(⇒ τ = r \times F\)

where τ is torque, r is the position vector of the point of application of force about the fixed axis.

\(⇒ τ = L \times F = LF sin 90=LF\)

• Moment of inertia I of the rod about one end is

\(⇒ I = \frac {ML^2}{3}\)​

Also, 

\(⇒ τ = I \alpha\)

\(⇒ LF = \frac {ML^2}{3} \times \alpha\)

\(⇒ \alpha= \frac {3F }{ML } \)

• Using the first equation of motion, 

\(⇒ ω = ω _0 + α t\)

\(⇒ ω = 0+ \frac {3F }{ML } t=\frac {3 Ft }{ML } \)

• Therefore option 3 is correct. 

The correct answer is option 2) i.e. 200 N

CONCEPT:

• Torque: Torque is the turning effect produced on an object due to a force acting on it at a distance from a fixed point. The object tends to turn about this fixed point.
  • It is defined mathematically as the cross product of force and the perpendicular distance to an axis of rotation.
  • Consider a fore F acting at a distance r from a fixed point O as shown.

The torque produced is given by:

τ = r × F = rFsinθ 

• The SI unit of torque is N-m.

CALCULATION:

• The opening of a nut using a spanner works on the principle of torque.
• When a force is applied through a spanner, the fixed point lies where the nut is positioned. As a result, a turning effect is produced on the nut which helps in its opening or tightening.
• The angle between force and moment arm is 90^∘ when maximum torque is produced.

Given that:

Length of the spanner = moment arm, r = 10 cm = 0.1 m

Force applied = 120 N

Torque = rF = 0.1 × 120 = 12 N-m

To open the nut using another spanner, the same torque has to be produced.

Lenght of the second spanner, r = 6 cm = 0.06 m

τ = rF

12 = 0.06 × F

⇒ F = 200 N

Concept:

Torque:

• The action of a force that acts on a point with some distance from the action of the force is called Torque. If a force F is applied at a distance r from the point of action, then torque τ is 

τ = Force × distance. 

• A common example of torque is applying force on the hinge of door and door rotates. 

• Torque is a vector quantity which is basically the cross product of force and distance from applied point to point of action.
• Torque plays the same role for a rotational motion as force plays for linear motion.
• Torque is required for a body to change its angular acceleration.
• So, the correct option is Force in linear motion.

Additional Information

Other analogs relating to Linear and translation motion are shown in the table.

The correct answer is option 4) i.e. 4

CONCEPT:

​Relationship between Torque and Moment of Inertia:

For the linear motion: From Netwon's 2nd law:

F = ma ⇒ a = \(\frac{F}{m}\) 

Where a is the acceleration.

For the circular motion: Consider an object of mass m moving in a circular path of radius r.

Angular displacement = r.dθ

Acceleration = \(\frac{dv}{dt} =r \frac{d(dθ)}{dt} = rα \)    

Torque = Force × perpendicular distance = F × r = ma × r = m(rα) × r = mr2α      ----(1)

We know, the moment of inertia, I = mr2      ----(2)

Substituting (2) in (1) we get,

Torque, τ = Iα 

CALCULATION:

Given that:

Torque, τ = 10⁵ N-m

Mass, m = 10 kg

The radius of gyration, r = 50 m

Moment of inertia, I = mr² = 10 × 50² = 25000 kg m²

Torque, τ = Iα 

⇒ Angular acceleration, α = \(\frac{\tau}{I}\) = \(\frac{10^5}{25000}\) = 4 rad/s²

CONCEPT:

The torque is defined as the force applied to the body which tends of rotating and it is written as;

\(\bar \tau = \bar r \times \bar F\)

Here r is the distance and F is the force.

CALCULATION:

Given: Force, \(\rm \overrightarrow F = 4\widehat i - 3\widehat j\)

The distance is written as;

\(\bar r_1 = 5 \hat i +5\sqrt 3 \hat j\)

and \(\bar r_2 = -5 \hat i +5\sqrt 3 \hat j\)

Now, the torque at P about point 'O'

\(\bar \tau_P = \bar r_1 \times \bar F\)

As we know, \(\rm \overrightarrow F = 4\widehat i - 3\widehat j\) and,  \( \bar r_1 =5 \hat i +5\sqrt 3 \hat j\), therefore,

\(\bar \tau_P=( 5 \hat i +5\sqrt 3 \hat j) \times (4 \hat i - 3\hat j) \)

⇒ \(\bar \tau_P =( 15 \hat i -20 \sqrt 3 \hat j) (-\hat k)\)

Similarly, at the position Q is written as;

\(\bar \tau_Q = \bar r_2 \times \bar F\)

\(\rm \overrightarrow F = 4\widehat i - 3\widehat j\) and,  \(\bar r_2 = -5 \hat i +5\sqrt 3 \hat j\), therefore,

\(\bar \tau_Q =( -5 \hat i +5\sqrt 3 \hat j) \times (4 \hat i - 3\hat j) \)

⇒ \(\bar \tau_Q =( -15 \hat i - 20 \sqrt 3 \hat j) (\hat k)\)

Hence, option 1) is the correct answer.

Concept:

• Torque is a force that tends to cause rotation.
• \(τ = \overrightarrow F \times \overrightarrow r\)
•  Torque is a vector quantity.
• Rational analogue: It is a force in a linear motion is torque.

where τ = torque, = radius of the rotatory motion, and \(\overrightarrow F\) = force applied

Explanation:

Torque is necessary for a body to do rotational motion.

Additional Information

• Moment of Inertia is the tendency of a body in rotational motion which opposes the change in its rotational motion due to external forces.
• Moment of Inertia behaves as angular mass and is called rotational inertia. 
• Angular momentum is defined as the product of the moment of inertia I and the angular velocity ω.
• L = Iω
• The radius of gyration is the distance from an axis of a body to the point in the body whose moment of inertia is equal to the moment of inertia of the entire body.

Concept:

Torque:

• Torque is a physical quantity that can cause an object to rotate about an axis.
• Force is what causes an object to accelerate in linear kinematics. Similarly, torque is what causes angular acceleration. Hence, torque can be defined as the rotational equivalent of linear force.
• Torque is a vector quantity.
• Its SI unit is N-m.
• If F force is acting at a distance r from the axis of rotation at an angle θ as shown in the figure, then the torque is given as,

​⇒ τ = F.r.sinθ

Couple:

• A pair of forces of equal magnitude but acting in opposite directions with different lines of action is known as a couple or torque.
• A couple produces rotation without translation.
• Examples:
  1. When we open the lid of a bottle by turning it, our fingers are applying a couple to the lid.

CONCEPT:

Torque: For a body rotating in a circular motion with a moment of inertia I and angular acceleration α 

the torque τ = Iα 

The magnetic moment of coil: The magnetic moment of a current-carrying coil M  = niA

Where, n = number of turns of the coil, A = area of the coil and i = current passing through the coil

EXPLANATION:

Given

n = 200, i = 1.6 A, r = 10 cm = 0.1 m, B = 0.72 T, I = 0.1 kg m2

Now, M = niA = 200×1.6×π(0.1)2 = 10.05 Am2

Where I and α are the moment of inertia and angular acceleration respectively

then the torque τ = Iα 

Again we know, τ = MB sinθ (M = magnetic moment)

So, Iα = MB sinθ

\(\begin{aligned} & \mathrm{I\omega} \frac{\mathrm{d\omega}}{\mathrm{d} \theta}=\mathrm{MB} \sin \theta \\ & \mathrm{I} \int_0^{\mathrm{\omega}} \mathrm{\omega d\omega}=\mathrm{MB} \int_{0^{\circ}}^{90^{\circ}} \sin \theta \mathrm{d} \theta \\ & \mathrm{I} \frac{\mathrm{\omega}^2}{2}=\mathrm{MB} \\ & \mathrm{\omega}=\sqrt{\frac{2 \mathrm{MB}}{\mathrm{I}}} \\ \end{aligned}\)

\(\mathrm{\omega}=\sqrt{\frac{ \mathrm{2×10.05×0.72 }}{\mathrm{0.1}}}\) = 12.03  rad/s

Hence the correct answer is option 1.

Calculation:

The torque is τ = r × F = [M1 L2 T-2]

The formula for the propagation of error in a product of quantities is:

Δτ / τ = (ΔM / M) + (2 * ΔL / L) + (2 * ΔT / T)

Where:

Δτ is the uncertainty in torque

τ is the measured torque

ΔM is the uncertainty in mass

M is the mass

ΔL is the uncertainty in length

L is the length

ΔT is the uncertainty in time

T is the time

Since each reference standard (mass, length, and time) has an uncertainty of 5%, we substitute ΔM / M = 5%, ΔL / L = 5%, and ΔT / T = 5% into the formula:

Δτ / τ = 5% + 2 * 5% + 2 * 5%

Δτ / τ = 5% + 10% + 10%

Δτ / τ = 25%

Thus, the net uncertainty in the measured torque is 25%. Therefore, the correct answer is:

Option 2: 25%

Concept:

Torque:

• Torque is the measure of the force that can cause an object to rotate about an axis.
• Force is what causes an object to accelerate in linear kinematics. Similarly, torque is what causes angular acceleration.
• Hence, torque can be defined as the rotational equivalent of linear force. 
• It is equal to the cross-product of the force and radius of rotatory motion.
• Formula, Torque, \(\vec \tau=\vec r\times \vec F\), where \(\vec \tau\) is the torque, \(\vec r\) is the radius of the rotatory motion and \(\vec F\) is the force applied.

Explanation:

Rational analog: It is a force in a linear motion that is torque.

Formula to calculate rational analog:

\(\vec \tau=\vec r\times \vec F\)
where \(\vec \tau\) is the torque, \(\vec r\) is the radius of the rotatory motion, and \(\vec F\) is the force applied.

• Force is necessary for a body to do translational motion.
• Similarly, it is torque that is required to rotate a body.
• If the net torque applied to the body about the axis of rotation is zero, then the body does not rotate.

Hence, the rotational analogue of force is torque.

Additional Information Acceleration:

• It is defined as the rate of change of velocity with respect to time.
• Formula, acceleration, \(a=\frac{dv}{dt}\)
• The SI unit of acceleration is m/s².

Moment of inertia:

• It is defined as the quantity expressed by the body resisting angular acceleration which is the sum of the product of the mass of every particle with its square of a distance from the axis of rotation. 
• Formula, I = mr², where m = mass, r =  distance from the axis of rotation. 
• The moment of Inertia is also known as the angular mass or rotational inertia. 
• The SI unit of moment of inertia is kg m2.

Pseudo force:

• Pseudo force comes into effect when the frame of reference has started acceleration compared to a non-accelerating frame.
• For example, if you consider a person standing at a bus stop watching an accelerating car, he infers that a force is exerted on the car and it is accelerating.