Question

A \(3.00-\mathrm{kg}\) broom is leaning against a coffee table. A woman lifts the broom handle with her arm fully stretched so that her hand is a distance of \(0.450 \mathrm{~m}\) from her shoulder. What torque is produced on her shoulder by the broom if her arm is at an angle of \(50.0^{\circ}\) below the horizontal? a) \(7.00 \mathrm{~N} \mathrm{~m}\) c) \(8.51 \mathrm{~N} \mathrm{~m}\) b) \(5.80 \mathrm{~N} \mathrm{~m}\) d) \(10.1 \mathrm{~N} \mathrm{~m}\)

Short answer

Question: A woman holds a 3.00 kg broom at an angle of 50.0° below the horizontal to clean her house. The distance between her hand and her shoulder (the axis of rotation) is 0.450 m. Calculate the torque produced on her shoulder. a) 4.52 N.m b) 8.85 N.m c) 9.59 N.m d) 10.1 N.m Answer: d) 10.1 N.m

Step-by-step solution

1. Identify the given information

The mass of the broom is \(3.00\,\mathrm{kg}\), the distance between the woman's hand and her shoulder is \(0.450\,\mathrm{m}\), and the angle between her arm and the horizontal plane is \(50.0^{\circ}\).

2. Calculate the weight of the broom

We can calculate the weight of the broom (force due to gravity) using the formula \(F_g = m \times g\), where \(m\) is the mass and \(g\) is the acceleration due to gravity \((9.81 \,\mathrm{m/s^2})\). \(F_g = 3.00\,\mathrm{kg} \times 9.81\,\mathrm{m/s^2} = 29.43\,\mathrm{N}\)

3. Calculate the vertical force component

The vertical component of the force can be found by multiplying the weight of the broom by the sine of the angle at which the arm is holding the broom (\(50.0^{\circ}\) below the horizontal). \(F_{vertical} = F_g \times \sin(50.0^{\circ}) = 29.43\,\mathrm{N} \times \sin(50.0^{\circ}) = 22.55\,\mathrm{N}\)

4. Calculate the torque

Finally, we can calculate the torque produced on the woman's shoulder using the formula \(\tau = r \times F_{vertical}\), where \(r\) is the distance between her hand and her shoulder. \(\tau = 0.450\,\mathrm{m} \times 22.55\,\mathrm{N} = 10.15\,\mathrm{N\,m}\)

5. Choose the correct answer

Comparing the calculated torque with the available choices, we see that the correct answer is closest to \(\boxed{10.1\,\mathrm{N\,m}}\) (option d).

Key concepts

Physics Problem Solving

Understanding physics involves recognizing the principles at play and systematically applying them to find a solution. In the scenario of the leaning broom, problem solving begins with identifying the known variables: the mass of the broom, the distance from the shoulder to the hand, and the angle of the arm. Once these variables are determined, we move on to apply principles of mechanics to find the force due to gravity (the broom's weight) and the torque. Each step logically follows from the previous, ensuring that all forces and angles are considered.

When students approach physics problems, it is crucial to break down complex questions into smaller, more manageable parts. This methodical approach not only makes it easier to solve the problem but also builds a strong foundation for understanding the underlying concepts. Notably, cross-checking the final solution with the given options is an integral step that reinforces the learning process and confirms the accuracy of the result.

Rotational Dynamics

Rotational dynamics is a fundamental concept in mechanics that deals with the motion of objects as they rotate. It requires a comprehensive understanding of torque, which is the rotational equivalent of force. In our broom example, torque is the pivotal factor exerted by the broom on the woman's shoulder, causing a rotational effect. The torque calculation involves both the magnitude of the force and the distance from the pivot point, which in this case is the shoulder joint.

To better visualize this, imagine trying to open a door. When you push near the hinge (pivot point), the door is difficult to move. However, when you push near the handle, which is farthest from the hinge, the door opens easily. This illustrates how the same force can exert different torques, depending on the distance from the pivot point. The broom problem likewise demonstrates this principle and highlights the importance of the radius (or distance) in calculating torque, thereby weaving the concept of rotational dynamics into the practical problem-solving scenario.

Trigonometry in Physics

Trigonometry, the branch of mathematics dealing with the relationships between the sides and angles of triangles, is widely used in physics, particularly when it comes to analyzing forces and motion. In our broom example, trigonometry helps us determine the component of force acting vertically, which is crucial for calculating torque. By using the sine function, a trigonometric function that relates a non-right angle of a triangle to the ratio of the opposite side over the hypotenuse, we can decompose the gravitational force acting on the broom into components aligned with and perpendicular to the arm.

One of the key aspects when applying trigonometry in physics is choosing the correct trigonometric function. For torque calculations, we are typically interested in the perpendicular distance, hence the use of the sine function in our problem. Mastery of trigonometry allows students to solve a wide array of physics problems and underscores its significance as a fundamental tool in the subject's problem-solving toolkit.