Q: Given the torque formula $$\tau = rF \sin \phi$$, what is the torque produced by a 50 N force applied at a distance of 0.3 m from the pivot at an angle of $$60^\circ to $$the lever arm?

$$\tau = 0.3 \times 50 \times \sin 60^\circ = 12.99 Nm$$

Q: Two masses, $$m_1 = 3$ kg at x_1$ = 0 m $$and $$m_2 = 7$ kg at x_2$ = 5 m$$, are connected by a light rod. What is the center of gravity of the system?

$$x_{cg} = \frac{0 \times 3 + 5 \times 7}{3 + 7} = 3.5 m$$

Q: A 0.80 m long ladder of mass 5 kg leans against a wall at an angle of $$50^\circ$$ with the floor. What is the torque due to its weight about the point where it touches the floor? (Take $$g = 9.8 m/s$^2)$$

$$\tau = 5 \times 9.8 \times 0.4 \times \sin 50^\circ = 15.0 Nm$$

Q: A uniform rod of length 2 m and mass 10 kg is supported at one end. A 20 kg mass is attached to the other end. Where is the center of gravity of the system from the supported end?

$$x_{cg} = \frac{10 \times 1 + 20 \times 2}{10 + 20} = 1.67 m$$

Question 1

Which of the following best describes the physical meaning of torque ($$	au) in $$the context of opening a door?

Accepted Answer

Torque is the product of the force applied and the distance from the pivot point to where the force is applied, considering only the component of force perpendicular to the lever arm.

Question 2

A mechanic uses two wrenches to loosen a bolt. Wrench A is 10 cm long, and Wrench B is 20 cm long. Both apply the same force perpendicular to the handle. Which wrench produces the greater torque, and by what factor?

Accepted Answer

Wrench B produces twice the torque of Wrench A.

Question 3

Given the torque formula $$	au = rF \sin \phi$$, what is the torque produced by a 50 N force applied at a distance of 0.3 m from the pivot at an angle of $$60^\circ to $$the lever arm?

Accepted Answer

$$	au = 0.3 	imes 50 	imes \sin 60^\circ = 12.99 Nm$$

Question 4

A uniform beam of length 4 m and mass 20 kg is supported at one end. Where is its center of gravity located?

Accepted Answer

At 2 m from the supported end

Question 5

Two masses, $$m_1 = 3$ kg at x_1$ = 0 m $$and $$m_2 = 7$ kg at x_2$ = 5 m$$, are connected by a light rod. What is the center of gravity of the system?

Accepted Answer

$$x_{cg} = \frac{0 	imes 3 + 5 	imes 7}{3 + 7} = 3.5 m$$

Question 6

A ladder of length $$L$$ leans against a frictionless wall, making an angle $$	heta$$ with the ground. What is the moment arm of the ladder's weight about the point where it touches the ground?

Accepted Answer

$$\frac{L}{2} \sin 	heta$$

Question 7

A 0.80 m long ladder of mass 5 kg leans against a wall at an angle of $$50^\circ$$ with the floor. What is the torque due to its weight about the point where it touches the floor? (Take $$g = 9.8 m/s$^2)$$

Accepted Answer

$$	au = 5 	imes 9.8 	imes 0.4 	imes \sin 50^\circ = 15.0 Nm$$

Question 8

Which of the following conditions must be satisfied for an object to be in static equilibrium?

Accepted Answer

Both the net force and the net torque on the object must be zero.

Question 9

A door is pushed at three different points: near the hinge, at the center, and at the edge farthest from the hinge. The same force is applied perpendicular to the door at each point. Rank the effectiveness of the force in opening the door from least to greatest.

Accepted Answer

Hinge \u003c Center \u003c Edge

Question 10

A gymnast balances horizontally on a beam. If her center of gravity is not directly above the support, what will happen?

Accepted Answer

She will rotate until her center of gravity is above the support.

Question 11

A uniform rod of length 2 m and mass 10 kg is supported at one end. A 20 kg mass is attached to the other end. Where is the center of gravity of the system from the supported end?

Accepted Answer

$$x_{cg} = \frac{10 	imes 1 + 20 	imes 2}{10 + 20} = 1.67 m$$

Question 12

If multiple forces act on an object, how is the net torque about a pivot calculated?

Accepted Answer

By summing the torques due to each force, taking into account their directions (signs).

Question 13

A lifebuoy is thrown to a swimmer. Its center of gravity is important because:

Accepted Answer

It is the point where the weight of the lifebuoy can be considered to act.

Question 14

A toy ladder leans against a frictionless wall and rests on a rough floor. To calculate the torque on the ladder about the point where it touches the floor due to the normal force by the wall, what is the moment arm?

Accepted Answer

$$L \sin 	heta$$

Question 15

Which of the following real-life scenarios best illustrates the concept of increasing torque by increasing the moment arm?

Accepted Answer

Using a longer wrench to loosen a tight bolt.

Question 16

A 1.00 m long beam of mass 10.0 kg is joined end-to-end with a 2.00 m long beam of mass 40.0 kg. How far from the left end of the left beam is the center of gravity of the system?

Accepted Answer

$$x_{cg} = \frac{10.0 	imes 0.5 + 40.0 	imes 2.0}{10.0 + 40.0} = 1.7 m$$

Torque and Center of Gravity: Concepts, Applications, and Equilibrium

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