4.2 Moment of a Force

In the previous section, we learned that a force can cause an object to move from one place to another. But forces can also cause objects to rotate. For example, when you open a door, you apply a force at the handle, and the door rotates around its hinges. This turning effect of a force is called the moment of force, or torque. Understanding torque is essential for analyzing rotational motion, just as force is essential for analyzing translational motion.

Definition

Moment of Force (or Torque): The turning effect produced in a body about a fixed point due to an applied force is called the moment of force or torque. It is a vector quantity with the SI unit of newton-metre (N·m).

Mathematically: τ = F × d where: - τ is the torque. - F is the applied force. - d is the moment arm (the perpendicular distance from the axis of rotation to the line of action of the force).

Main Idea

The main idea is that force can cause rotation, and the effectiveness of a force in causing rotation depends on both the magnitude of the force and the distance from the pivot. A larger force or a larger moment arm produces a larger torque. This is why it is easier to open a door by pushing near the handle (far from the hinges) than by pushing near the hinges. Torque is the rotational analogue of force.

Core Concept

The Formula for Torque The magnitude of torque is given by: τ = F × d where: - F is the magnitude of the force. - d is the moment arm (the perpendicular distance from the axis of rotation to the line of action of the force).

The text states: "The moment of force or torque τ is equal to the magnitude of the force 'F' multiplied by the perpendicular distance from the axis of rotation to the line of action of force 'd'."

Moment Arm The moment arm is the perpendicular distance from the axis of rotation (pivot) to the line of action of the force. It is a crucial factor in determining torque. The text explains that if the force is not perpendicular to the moment arm, you must extend the line of action of the force and take the moment arm as the perpendicular distance from the axis of rotation to that line.

• To increase torque, you can increase the force.
• To increase torque, you can increase the moment arm (e.g., use a longer wrench).

• Clockwise Moment: The force tends to rotate the body in a clockwise direction. By convention, this is taken as negative.
• Anticlockwise Moment: The force tends to rotate the body in an anticlockwise direction. By convention, this is taken as positive.

• d_A = 0.40 m\): \(τ_A = 0.40 × 60 = 24.0 N·m
• d_B = 0.20 m\): \(τ_B = 0.20 × 60 = 12.0 N·m
• d_C = 0.0 m\): \(τ_C = 0 × 60 = 0 N·m

This shows that the torque is maximum when the force is applied farthest from the pivot and zero when applied at the pivot.

Detailed Explanation

Torque is the rotational analogue of force. Just as force causes changes in translational motion (acceleration), torque causes changes in rotational motion (angular acceleration).

The Role of the Moment Arm The moment arm is the perpendicular distance from the pivot to the line of action of the force. It is not necessarily the distance from the pivot to the point of application of the force. If the force is applied at an angle, the moment arm is the perpendicular component of that distance.

For example, if you pull a door handle at an angle, the effective moment arm is smaller than if you pull perpendicular to the door. This is why it's easier to push a door perpendicular to its surface.

Torque and Newton's Laws for Rotation The text states: "Just like force causes change in motion, moment of force causes change in rotation. This means that moment of force play the same role in rotational motion as force in translational motion."

• First Law (Rotational): An object at rest tends to remain at rest, and an object that is rotating tends to spin with a constant angular velocity, unless acted on by a nonzero net external torque.
• Second Law (Rotational): The net torque on an object is equal to the moment of inertia times the angular acceleration (τ = Iα).
• Third Law (Rotational): For every torque, there is an equal and opposite torque.

• Wrenches and spanners: A longer wrench provides a larger moment arm, making it easier to loosen a tight nut.
• Door handles: Placed far from the hinges to provide a larger moment arm, making it easier to open the door.
• Steering wheels: The larger the steering wheel, the easier it is to turn.
• Levers, pulleys, and gears: All rely on the principle of torque.

Intermediate Level

In physics, torque is the rotational equivalent of force. It is defined as the cross product of the position vector (from the pivot to the point of application) and the force vector: τ = r × F. The magnitude is τ = Fdsinθ, where θ is the angle between the force and the moment arm. The moment arm is d = rsinθ, which is the perpendicular distance from the pivot to the line of action of the force.

The direction of torque is given by the right-hand rule. If you curl the fingers of your right hand in the direction of rotation, your thumb points in the direction of the torque vector (perpendicular to the plane of rotation).

Torque is essential for understanding rotational equilibrium. For an object to be in rotational equilibrium, the net torque acting on it must be zero. This means the sum of clockwise torques must equal the sum of anticlockwise torques (the principle of moments).

Important Points

• - Torque (Moment of Force): The turning effect of a force.
• - Formula: τ = F × d (magnitude).
• - Moment Arm (d): Perpendicular distance from pivot to line of action.
• - Unit: Newton-metre (N·m).
• - Vector Quantity: Has magnitude and direction (clockwise or anticlockwise).
• - Sign Convention: Anticlockwise = positive, Clockwise = negative.
• - Increasing Torque: Increase force or increase moment arm.

Remember

• - Torque = Force × Moment Arm.
• - Formula: τ = F × d.
• - Moment Arm: Perpendicular distance from pivot.
• - Unit: N·m.
• - Clockwise: Negative. Anticlockwise: Positive.
• - Longer wrench = more torque.

Common Mistakes

• - Using the wrong distance: The moment arm is the perpendicular distance from the pivot to the line of action, not necessarily the distance from the pivot to the point of application.
• - Forgetting the direction: Torque is a vector. Clockwise and anticlockwise torques have opposite signs.
• - Confusing torque with force: Torque is the turning effect; force is the push or pull.

Quick Revision

The moment of force, or torque, is the turning effect produced by a force about a pivot point. It is calculated as the product of the force and the moment arm (the perpendicular distance from the pivot to the line of action of the force). Torque is a vector quantity with the SI unit of newton-metre (N·m). It can cause clockwise or anticlockwise rotation, and its magnitude increases with either a larger force or a larger moment arm.

60-Second Revision

• Formula: τ = F × d.
• d: Perpendicular distance from pivot to line of action.
• Unit: N·m.
• Sign: Anticlockwise (+), Clockwise (-).
• Increasing torque: Increase force or moment arm.
• Example: Longer wrench for more torque.

Frequently Asked Questions (FAQs)

Q1. What is the moment of force? Answer: The moment of force, or torque, is the turning effect produced in a body about a fixed point due to an applied force. It is the product of the force and the moment arm.

• Q2. What is the formula for torque?
• Answer: The formula for torque is:
• τ = F × d
• or in magnitude, τ = F × d, where d is the moment arm.

• Q3. What is the moment arm?
• Answer: The moment arm is the perpendicular distance from the axis of rotation (pivot) to the line of action of the force.

• Q4. What is the SI unit of torque?
• Answer: The SI unit of torque is the newton-metre (N·m).

• Q5. Is torque a scalar or a vector?
• Answer: Torque is a vector quantity. It has both magnitude and direction (clockwise or anticlockwise).

Q6. How can you increase the torque for a given force? Answer: You can increase the torque by increasing the moment arm (the perpendicular distance from the pivot to the line of action of the force). For example, using a longer wrench.

• Q7. What is the sign convention for torque?
• Answer: By convention, anticlockwise torque is taken as positive, and clockwise torque is taken as negative.

Q8. Why is it easier to open a door using the handle rather than pushing near the hinges? Answer: The handle is farther from the hinges (the pivot), so it provides a larger moment arm. A larger moment arm produces a larger torque for the same applied force, making it easier to open the door.

Multiple Choice Questions (MCQs)

• Q1. The turning effect of a force is called:
• A. Force
• B. Momentum
• C. Torque
• D. Inertia

• Correct Answer: C
• Explanation: The text defines the turning effect of a force as the moment of force or torque.

• Q2. The formula for torque is:
• A. τ = Fd
• B. τ = F × d
• C. τ = F + d
• D. τ = dF

• Correct Answer: B
• Explanation: The text states that torque is equal to the magnitude of the force multiplied by the moment arm, τ = F × d.

• Q3. The perpendicular distance from the axis of rotation to the line of action of the force is called the:
• A. Force
• B. Moment arm
• C. Torque
• D. Line of action

• Correct Answer: B
• Explanation: The text defines the perpendicular distance 'd' as the moment arm.

• Q4. The SI unit of torque is:
• A. Newton (N)
• B. Joule (J)
• C. Newton-metre (N·m)
• D. Watt (W)

• Correct Answer: C
• Explanation: The text states that the SI unit of moment of force or torque is N·m.

• Q5. Torque is a:
• A. Scalar quantity
• B. Vector quantity
• C. Base quantity
• D. Derived quantity

• Correct Answer: B
• Explanation: The text states that moment of force or torque is a vector quantity.

• Q6. By convention, anticlockwise torque is taken as:
• A. Positive
• B. Negative
• C. Zero
• D. Infinite

• Correct Answer: A
• Explanation: The text states that conventionally, anticlockwise moment is taken as positive.

• Q7. By convention, clockwise torque is taken as:
• A. Positive
• B. Negative
• C. Zero
• D. Infinite

• Correct Answer: B
• Explanation: The text states that clockwise moment is taken as negative.

• Q8. To increase the torque for a given force, you should:
• A. Decrease the moment arm
• B. Increase the moment arm
• C. Decrease the force
• D. Apply the force at the pivot

• Correct Answer: B
• Explanation: The text states that increasing the moment arm 'd' increases the moment of force.

• Q9. If a force is applied at the pivot point, the torque is:
• A. Maximum
• B. Minimum
• C. Zero
• D. Infinite

• Correct Answer: C
• Explanation: If the force is applied at the pivot, the moment arm is zero, so the torque is zero.

• Q10. A force of 60 N is applied at a distance of 0.40 m from the pivot. The torque is:
• A. 24.0 N·m
• B. 12.0 N·m
• C. 6.0 N·m
• D. 0 N·m

• Correct Answer: A
• Explanation: τ = F × d = 60 × 0.40 = 24.0 N·m, as calculated in the text.

Frequently Asked Questions (FAQs)

Q1. What is the difference between torque and force? Answer: Force is a push or pull that can cause translational motion. Torque is the turning effect of a force that can cause rotational motion. Torque depends on both the force and the moment arm.

Q2. Can a force produce torque without producing translation? Answer: Yes. If two equal and opposite forces act at different points on an object, they can produce a net torque (a couple) without producing a net translational force.

Q3. What is the principle of moments? Answer: The principle of moments states that for an object in equilibrium, the sum of clockwise moments about a pivot is equal to the sum of anticlockwise moments about the same pivot.

Q4. How is torque related to angular acceleration? Answer: According to Newton's second law for rotation, the net torque on an object is equal to its moment of inertia times its angular acceleration (τ = Iα).

Q5. What is the direction of the torque vector? Answer: The direction of the torque vector is perpendicular to the plane of rotation, determined by the right-hand rule. If you curl the fingers of your right hand in the direction of rotation, your thumb points in the direction of the torque.

Q6. Why is a longer wrench more effective for loosening a bolt? Answer: A longer wrench provides a larger moment arm. For the same applied force, a larger moment arm produces a larger torque, making it easier to loosen the bolt.

Q7. What happens to the torque if the force is applied at an angle? Answer: If the force is applied at an angle, the moment arm is the perpendicular distance from the pivot to the line of action of the force. The torque is calculated using this perpendicular distance. The torque is maximum when the force is perpendicular to the moment arm.

You: