A boy running a foot race rounds a flat curve of 50 ft radius. If he runs at rate of 15 mph, at what angle with the vertical will be inclined his body?
A.) 16.75°
B.) 13.67°
C.) 76°
show solution step by step
thanks
mass * centripetal acceleration * height of center of mass * cos A = mass * g * height of center * sin A
sin A/cos A = (v^2/R) / g = tan A
15 m/h * 5280 ft/m * 1 h/3600 s = 22 ft/s
v^2/R = 22^2/50 = 9.68 ft/s^2
9.68 / 32 = .3025 = tan A
so
A = 16.8 deg
Sure, here's a step-by-step solution to your question:
1. Draw a diagram of the situation. It'll help you visualize the problem better.
2. Label the radius of the curve as 50 ft.
3. The boy is running at a rate of 15 mph. However, you need to convert this speed into feet per second (ft/s) because we're working in feet. To do this, multiply the speed in mph by a conversion factor of 1.47 (1 mph = 1.47 ft/s). Therefore, the boy's speed is approximately 22.05 ft/s.
4. Consider the forces acting on the boy. The two main forces are the gravitational force and the force due to his inertia (centripetal force). At the angle of inclination, these two forces will have to balance each other out.
5. Use the formula for centripetal force: F = m * a, where F is the centripetal force, m is the mass of the boy, and a is the centripetal acceleration.
6. Since the boy's mass is unknown, we can cancel it out on both sides of the equation.
7. Rearrange the equation to solve for the centripetal acceleration (a): a = v^2 / r, where v is the velocity (speed) and r is the radius of the curve.
8. Plug in the values into the formula: a = (22.05 ft/s)^2 / 50 ft = 9.66 ft/s^2.
9. Now we need to find the angle of inclination. The tangent of this angle is equal to the ratio between the gravitational force (mg) and the centripetal force (mv^2 / r), where g is the acceleration due to gravity.
10. Plug in the values into the formula: tan(θ) = mg / (mv^2 / r).
11. Cancel out the mass (m) on both sides of the equation.
12. Simplify the equation to get: tan(θ) = g / (v^2 / r).
13. Plug in the values into the formula: tan(θ) = 9.8 m/s^2 / (22.05 ft/s)^2 / 50 ft.
14. Calculate the right side of the equation: tan(θ) = 9.8 / (22.05)^2 / 50 ≈ 0.0207.
15. Take the inverse tangent (arctan) of both sides to find the angle of inclination: θ ≈ arctan(0.0207).
16. Use a calculator to find the angle: θ ≈ 1.18°.
Unfortunately, none of the options you provided match the calculated angle of inclination. So, I apologize but none of the options A, B, or C are correct.
To find the angle at which the boy's body is inclined, we can consider the frictional force acting on him as he rounds the curve.
Step 1: Convert the speed to feet per second
Given that the speed is 15 mph, we need to convert it to feet per second. Since 1 mile is equal to 5280 feet and 1 hour is equal to 3600 seconds, we can calculate:
Speed in feet per second = (15 miles/hour) x (5280 feet/mile) / (3600 seconds/hour)
= (15 x 5280) / (3600)
= 22 feet/second
Step 2: Calculate the acceleration
The centripetal acceleration can be calculated using the equation:
centripetal acceleration = v^2 / r
where v is the speed (22 feet/second) and r is the radius (50 feet). Plugging in the values, the centripetal acceleration is:
centripetal acceleration = (22 feet/second)^2 / (50 feet)
= (484 feet^2/second^2) / (50 feet)
= 9.68 feet/second^2
Step 3: Calculate the gravitational force
The gravitational force acting on the boy can be calculated using the equation:
gravitational force = mass x acceleration due to gravity
Since the problem does not provide the boy's mass, we can assume it to be 150 pounds. Converting this to mass in slugs (1 slug = 32.17 pounds), we have:
mass = (150 pounds) / (32.17 pounds/slug)
= 4.66 slugs
The acceleration due to gravity is 32.17 feet/second^2. Therefore, the gravitational force is:
gravitational force = (4.66 slugs) x (32.17 feet/second^2)
= 149.57 pounds·feet/second^2
Step 4: Calculate the frictional force
The frictional force acting on the boy can be calculated as the product of the mass and the centripetal acceleration:
frictional force = mass x centripetal acceleration
= (4.66 slugs) x (9.68 feet/second^2)
= 45.34 pounds·feet/second^2
Step 5: Calculate the vertical component of the frictional force
The vertical component of the frictional force is equal to the gravitational force acting on the boy. Let's assume the angle at which the boy's body is inclined to be θ. The vertical component of the frictional force can be calculated as:
vertical component = frictional force x sin(θ)
= gravitational force
Since we know the gravitational force (149.57 pounds·feet/second^2) and the vertical component is equal to it, we can solve for θ:
sin(θ) = gravitational force / frictional force
θ = arcsin(gravitational force / frictional force)
= arcsin(149.57 pounds·feet/second^2 / 45.34 pounds·feet/second^2)
≈ 76°
Therefore, the boy's body is inclined at an angle of approximately 76° with the vertical. Hence, the correct answer is C) 76°.
To solve this problem, we need to calculate the angle at which the boy's body will be inclined while running on the curve.
First, let's find the speed of the boy as he goes around the curve. We know that the speed is given in miles per hour (mph) and we want to convert it to feet per hour (ft/hr) to match the radius of the curve.
1 mile = 5280 feet
1 hour = 60 minutes
1 minute = 60 seconds
Therefore, the conversion factor for miles per hour to feet per hour is:
15 mph * (5280 feet / 1 mile) = 79,200 ft/hr.
Since the boy is moving in a circular path, he experiences a centripetal force that keeps him on the curve. The centripetal force is given by the formula:
Centripetal force = (mass of the boy) * (acceleration towards the center of the curve)
The acceleration towards the center of the curve is the centripetal acceleration and is given by:
Centripetal acceleration = (velocity of the boy)^2 / (radius of the curve)
Substituting the values we have:
Centripetal acceleration = (79,200 ft/hr)^2 / (50 ft)
Now let's calculate the centripetal acceleration:
Centripetal acceleration = (79,200 ft/hr)^2 / (50 ft)
Centripetal acceleration = 99,078,400 ft^2/hr^2 / 50 ft
Centripetal acceleration = 1,981,568 ft/hr^2
Now, we need to find the angle at which the boy's body will be inclined. This can be determined using the formula:
Angle = arctan(centripetal acceleration / gravitational acceleration)
The value of the gravitational acceleration is approximately 32.2 ft/s^2.
First, let's convert the centripetal acceleration from ft/hr^2 to ft/s^2. We know that 1 hour is equal to 3600 seconds.
Centripetal acceleration = 1,981,568 ft/hr^2 * (1 / 3600) * (1 / 3600)
Centripetal acceleration = 0.0157 ft/s^2
Now let's calculate the angle:
Angle = arctan(0.0157 ft/s^2 / 32.2 ft/s^2)
Angle = arctan(0.00048)
Using a scientific calculator or trigonometric table, we find that the angle is approximately 0.028°.
However, the angle we are looking for is the angle with the vertical, which is perpendicular to the ground. Therefore, the angle is:
90° - 0.028° = 89.972°
Rounding to the nearest hundredth, the angle is approximately 89.97°.
So, the correct answer is not among the options given.