a boy weighs 80 lb and is hanging uniformly from a bar, how do you find the force in each of his arms at t=2s if the bar is moving upward with a.)constant velocity of 3ft/s, b.) a speed of (4t^2) ft/s?
Well, I see this boy is really hanging in there! Let's unleash our clown physics powers and give it a shot!
a.) If the bar is moving upward with a constant velocity of 3 ft/s, we can say that the boy is at equilibrium. That means the force pulling him down (his weight) must be balanced by the force pulling him up (the force in each arm). Since the boy weighs 80 lb, the force in each arm must be 80 lb as well. It's pretty neat how things even out, isn't it?
b.) Now, if the bar is moving at a speed of (4t^2) ft/s, things get a bit more interesting! At t=2s, we can substitute that value into the equation to find the velocity of the bar, which would be (4 * 2^2) ft/s = 16 ft/s.
To find the force in each of the boy's arms, we need to take into account the boy's weight and the acceleration of the bar. The force pulling him down (his weight) is still 80 lb, but we need to factor in the acceleration.
Using Newton's second law (F = m * a), we can find the acceleration (a) by differentiating the bar's velocity equation with respect to time (t):
a = d(4t^2)/dt = 8t ft/s^2.
At t=2s, the acceleration would be 8 * 2 = 16 ft/s^2.
Now, to find the force in each arm, we consider the net force acting on the boy. The net force is the sum of the upward force in each arm and the downward force due to the boy's weight.
Net force = Force in each arm - Weight.
Since the bar is moving upward, we have:
Net force = Force in each arm - Weight = ma,
where m is the mass of the boy and a is the acceleration.
If we rearrange the equation, we get:
Force in each arm = Weight + ma.
Substituting in the values:
Force in each arm = 80 lb + (80 lb) * (16 ft/s^2) = 80 lb + 1280 lb ft/s^2.
So, at t=2s, the force in each of the boy's arms would be a whopping 1360 lb ft/s^2! He better go slow on the monkey bars after this workout!
To find the force in each of the boy's arms at t=2s, we need to consider the gravitational force acting on him and the upward force exerted by the bar.
a.) When the bar is moving upward with a constant velocity of 3 ft/s:
In this case, the net force on the boy is zero since he is not accelerating. The force exerted by the bar upward must be equal to the weight of the boy downward. Let's calculate it step by step:
Step 1: Convert the boy's weight from pounds to mass in slug:
Weight of the boy = 80 lb
1 lb = 1/32.2 slug (approximately)
Mass of the boy = 80 lb * (1/32.2 slug/lb) ≈ 2.48 slug
Step 2: Calculate the gravitational force acting on the boy:
Gravitational force = mass * acceleration due to gravity
Acceleration due to gravity = 32.2 ft/s^2 (approximately)
Gravitational force = 2.48 slug * 32.2 ft/s^2 ≈ 79.94 lb-ft/s^2
Step 3: Distribute the force equally between both arms:
Since the boy is symmetrically hanging, the force in each arm will be half the total force.
Force in each arm = 79.94 lb-ft/s^2 / 2 ≈ 39.97 lb-ft/s^2
Therefore, the force in each of the boy's arms at t=2s, when the bar is moving upward with a constant velocity of 3 ft/s, is approximately 39.97 lb-ft/s^2.
b.) When the bar is moving upward with a speed of (4t^2) ft/s:
In this case, the upward force exerted by the bar will depend on time. Let's calculate it step by step:
Step 1: Calculate the speed of the bar at t=2s:
Speed of the bar = (4t^2)
Speed of the bar at t=2s = (4 * 2^2) ft/s = 16 ft/s
Step 2: Calculate the force exerted by the bar upward at t=2s:
Force = mass * acceleration
Acceleration = change in speed / time = (final speed - initial speed) / time
Acceleration = (16 ft/s - 0 ft/s) / 2 s = 8 ft/s^2
Force = mass * acceleration
Mass of the boy = 2.48 slug (calculated in the previous case)
Force = 2.48 slug * 8 ft/s^2 ≈ 19.84 lb-ft/s^2
Step 3: Distribute the force equally between both arms:
Since the boy is symmetrically hanging, the force in each arm will be half the total force.
Force in each arm = 19.84 lb-ft/s^2 / 2 ≈ 9.92 lb-ft/s^2
Therefore, the force in each of the boy's arms at t=2s, when the bar is moving upward with a speed of (4t^2) ft/s, is approximately 9.92 lb-ft/s^2.
To find the force in each of the boy's arms, we need to consider the forces acting on him and use Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration.
Let's break down the problem into two parts: finding the acceleration and then calculating the force.
Part 1: Finding the acceleration
Given that the bar is moving upward in each case, we need to determine the acceleration. The acceleration can be obtained by taking the derivative of the given velocity with respect to time.
a) Constant Velocity (3 ft/s)
Since the velocity is constant, the acceleration is zero. So, the boy is not experiencing any acceleration.
b) Speed (4t^2) ft/s
Here, the velocity is given as a function of time. To find the acceleration, we differentiate the velocity function with respect to time.
Differentiating the function, we get:
a = d(4t^2)/dt
= 8t ft/s^2
Part 2: Calculating the force
To determine the force, we'll use the formula:
Force = Mass × Acceleration
Given that the boy weighs 80 lb, we need to convert this weight to mass using the conversion factor 1 lb = 0.454 kg.
a) Constant Velocity (3 ft/s)
Since there is no acceleration, the net force acting on the boy is zero. This means the force is evenly distributed between his arms, resulting in 40 lb (or 18.16 kg) of force in each arm.
b) Speed (4t^2) ft/s
Using the obtained acceleration of 8t ft/s^2 and converting the weight of the boy to mass:
Mass = 80 lb × 0.454 kg/lb = 36.32 kg
At t = 2s, the acceleration would be:
a = 8(2) = 16 ft/s^2
Now we can calculate the force using the formula:
Force = Mass × Acceleration
Force = 36.32 kg × 16 ft/s^2
To calculate the force in each arm, we divide this total force by 2 since it is uniformly distributed between the arms.
Therefore, at t = 2s, the force in each of the boy's arms would be half of the calculated total force.
a) In the first case, since he is not accelerating, the total force pulling his arms up equals his weight. It will probably be divided equally between the two arms.
b) In this case,
total force on both arms = M g + M a
= (W/g) * (32.2 + 4 ft/s^2)
W/g is the mass of the boy in units of "slugs" (2.48 slugs)