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Chapter 3: Problem 329

The 180 -lb exerciser is beginning to execute a bicep curl. When in the position shown with his right elbow fixed, he causes the 20 -lb cylinder to accelerate upward at the rate \(g / 4 .\) Neglect the effects of the mass of his lower arm and estimate the normal reaction forces at \(A\) and \(B .\) Friction is sufficient to prevent slipping.

Short Answer

Expert verified
Calculate normal forces using equilibrium: \( N_A + N_B = 25 \text{ lb} \).

Step by step solution

01

Understand the problem

We have a system involving a bicep curl with a weight. The exerciser is accelerating the weight upward. Our goal is to find the normal reaction forces at points \( A \) and \( B \) where the arm is supported.
02

Identify the forces

The forces involved include the weight of the cylinder \((20 \text{ lb})\), the effective force due to acceleration, and the weight of the exerciser's arm. Friction prevents slipping, indicating a static situation where normal forces exist.
03

Calculate the effective force due to acceleration

The acceleration is given as \( \frac{g}{4} \). The effective gravitational force on the 20 lb weight is \( F = m \times a = 20 \times \left( \frac{g}{4} \right) = 5 \text{ lb} \).
04

Set up equations for equilibrium

Set up equilibrium equations for forces in the vertical direction. Let \( N_A \) and \( N_B \) be the normal reactions at points \( A \) and \( B \). Due to equilibrium: \( N_A + N_B - 20 - 5 = 0 \). Solve for one variable in terms of the other.
05

Calculate normal forces

Using the equilibrium equation, \( N_A + N_B = 25 \text{ lb} \). Additional equations or assumptions about the distribution of load between \( A \) and \( B \) may solve the exact values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Forces
Normal forces are the support forces exerted by a surface perpendicular to an object placed on it. In this exercise, we are interested in finding the normal forces at points \(A\) and \(B\), which offer support during a bicep curl. These forces are crucial as they balance out other forces acting on the arm and weight, ensuring stability during the movement.

  • These points are where the arms contact a surface, which could be imagined as forming a simple system of pulleys and levers.
  • Despite not being vertical, understanding these forces' components allow us to describe the equilibrium state of the exerciser's arm.
To solve real-world dynamics problems, comprehending how normal forces act to prevent movement and increase stability is essential. In this case, normal forces prevent the weight from causing the exerciser's arm to slip downwards.
Equilibrium Equations
In dynamics, equilibrium equations help us understand how forces balance in a system. When an object is stationary or moving at a constant velocity, the sum of the forces鈥攁nd sometimes the sum of torques鈥攁cting on it must equal zero. This exercise is a perfect example of applying equilibrium to a physical scenario.

  • We sum forces vertically to find the balance equation for the system: \( N_A + N_B - 20\, \text{lb} - 5\, \text{lb} = 0 \).
  • Here, \(20\, \text{lb}\) is the weight of the cylinder, and \(5\, \text{lb}\) is the additional force due to acceleration.
By deducting this combined force from the sum, we can find the total normal force required to maintain equilibrium. This is the core of solving such problems鈥攅nsuring that all forces, including unknowns like \(N_A\) and \(N_B\), fit into an equation that sums to zero.
Bicep Curl Mechanics
The mechanics of a bicep curl involves raising a weight by contracting the bicep muscle. In this exercise, dynamics come into play, as the cylinder is accelerated upward. Let's break down how this exercise works:

  • A force is exerted by the bicep to lift and accelerate the weight, overcoming the gravitational force acting downward.
  • Additionally, the body's segments play a role; the forearm's orientation changes the moment arm, affecting how the weight behaves in motion.
The integrity of this maneuver depends on the coordination of muscle forces and equilibrium. The challenge lies in maintaining enough normal force at support points \(A\) and \(B\) to counteract the downward pull of weight and the added inertia from acceleration.
Acceleration Effects
When a force causes an object to accelerate, it changes how other forces interact and balance in a system. In our scenario, the cylinder is accelerated upward at a rate of \( \frac{g}{4} \), adding an effective force to consider.

  • Acceleration must be included in the balance equations to reflect the true forces at play. The effective force is calculated by multiplying the object's mass by its acceleration.
  • In this case, \(20\, \text{lb} \times \frac{g}{4} = 5\, \text{lb}\), is the additional upward force required to accelerate the weight.
Understanding how acceleration affects a dynamic system helps us see why even small forces can significantly influence equilibrium. Especially in workouts, recognizing these effects allows for optimizing performance and preventing injury.

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Most popular questions from this chapter

The aircraft carrier is moving at a constant speed and launches a jet plane with a mass of \(3 \mathrm{Mg}\) in a distance of \(75 \mathrm{m}\) along the deck by means of a steam-driven catapult. If the plane leaves the deck with a velocity of \(240 \mathrm{km} / \mathrm{h}\) relative to the carrier and if the jet thrust is constant at \(22 \mathrm{kN}\) during takeoff, compute the constant force \(P\) exerted by the catapult on the airplane during the \(75-\mathrm{m}\) travel of the launch carriage.

The pilot of a 90,000 -lb airplane which is originally flying horizontally at a speed of \(400 \mathrm{mi} / \mathrm{hr}\) cuts off all engine power and enters a \(5^{\circ}\) glide path as shown. After 120 seconds the airspeed is \(360 \mathrm{mi} / \mathrm{hr} .\) Calculate the time-average drag force \(D(\) air resistance to motion along the flight path)

The slotted arm \(O A\) rotates about a fixed axis through \(O\). At the instant under consideration, \(\theta=\) \(30^{\circ}, \dot{\theta}=45 \operatorname{deg} / \mathrm{s},\) and \(\ddot{\theta}=20 \mathrm{deg} / \mathrm{s}^{2} .\) Determine the forces applied by both arm \(O A\) and the sides of the slot to the 0.2 -kg slider \(B\). Neglect all friction, and let \(L=0.6 \mathrm{m} .\) The motion occurs in a vertical plane.

The cars of an amusement-park ride have a speed \(v_{1}=90 \mathrm{km} / \mathrm{h}\) at the lowest part of the track. Determine their speed \(v_{2}\) at the highest part of the track. Neglect energy loss due to friction. (Caution: Give careful thought to the change in potential energy of the system of cars.)

A 175 -lb pole vaulter carrying a uniform \(16-\mathrm{ft}\) 10-lb pole approaches the jump with a velocity \(v\) and manages to barely clear the bar set at a height of \(18 \mathrm{ft}\). As he clears the bar, his velocity and that of the pole are essentially zero. Calculate the minimum possible value of \(v\) required for him to make the jump. Both the horizontal pole and the center of gravity of the vaulter are 42 in. above the ground during the approach.
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