Just an instant before reaching the very bottom of a semicircular section of a roller coaster ride, the automatic emergency brake inadvertently goes on. Assume the car has a total mass of 850 kg, the radius of that section of the track is 53.0 m, and the car entered the bottom after descending vertically (from rest) 29.0 m on a frictionless straight incline.
If the braking force is a steady 1800 N, determine the magnitude of the car's centripetal acceleration.
Determine the normal force of the track on the car.
Determine the magnitude of the tangential acceleration of the car.
Determine the total acceleration of the car.
Help me solve this problem.
To solve this problem, we can use the principles of kinetic energy and circular motion. Let's break down each step:
1. Determining the magnitude of the car's centripetal acceleration:
The centripetal acceleration can be found using the formula: ac = v^2 / r, where v is the velocity and r is the radius. Initially, the car is at rest, so we need to find the velocity when it reaches the bottom of the incline.
Using the principle of conservation of energy, we can equate the potential energy at the top of the incline to the kinetic energy at the bottom: mgh = (1/2)mv^2.
This equation simplifies to: gh = v^2 / 2, where g is the acceleration due to gravity.
Substituting the given values: 9.8 m/s^2 * 29 m = v^2 / 2, we can solve for v^2:
v^2 = 9.8 m/s^2 * 29 m * 2.
Using the value of v^2, we can calculate the centripetal acceleration:
ac = v^2 / r = (9.8 m/s^2 * 29 m * 2) / 53 m.
2. Determining the normal force of the track on the car:
The normal force (Fn) is the force exerted by a surface perpendicular to it. At the bottom of the roller coaster, the normal force is the force that balances the weight of the car.
Fn = mg + Fbrake, where mg is the weight and Fbrake is the force applied by the brake.
Substituting the given values: Fn = 850 kg * 9.8 m/s^2 + 1800 N.
3. Determining the magnitude of the tangential acceleration:
The tangential acceleration is the component of acceleration tangent to the circular path. Since the car descended vertically before entering the semicircular section, the tangential acceleration can be found using the equation: at = Fnet / m, where Fnet is the net force acting on the car.
Initially, the net force is the force due to gravity, so at = mg / m = 9.8 m/s^2.
4. Determining the total acceleration of the car:
The total acceleration (a) is the combination of centripetal and tangential acceleration. We can use the Pythagorean theorem to calculate it:
a = sqrt(ac^2 + at^2).
Now, let's substitute the values into the equations and solve them:
1. ac = (9.8 m/s^2 * 29 m * 2) / 53 m.
2. Fn = 850 kg * 9.8 m/s^2 + 1800 N.
3. at = 9.8 m/s^2.
4. a = sqrt(ac^2 + at^2), using the value obtained in step 1.
Calculating these values will give you the answers to the problem.
To solve this problem, we need to consider the different forces acting on the car at the bottom of the roller coaster. Let's break it down step by step:
1. Finding the magnitude of the car's centripetal acceleration:
The centripetal acceleration is caused by the net force acting towards the center of the circular path. In this case, the net force is the combination of gravitational force and the braking force. We can find the net force using the equation:
Net Force = Frictional force - Weight
Weight (W) = mass (m) * gravitational acceleration (g)
The frictional force is equal to the braking force (1800 N) since there is no skidding or slipping occurring on the track.
Net Force = 1800 N - (mass * gravitational acceleration)
Next, we need to find the mass of the car. We are given the total mass of the car, which is 850 kg.
Now, we can calculate the net force:
Net Force = 1800 N - (850 kg * 9.8 m/s^2) = ...
Simplifying and calculating this will give you the value of the net force. Once you have the net force, you can use the following equation to find the centripetal acceleration:
Centripetal acceleration = Net Force / Mass
Plugging in the values, you'll be able to calculate the magnitude of the car's centripetal acceleration.
2. Finding the normal force of the track on the car:
At the bottom of the roller coaster loop, the normal force and weight of the car are in opposite directions. This is because the normal force acts perpendicular to the track and opposes the weight of the car. The sum of the normal force and weight should equal the centripetal force.
Centripetal force = Normal force + Weight
Using this equation, you can solve for the normal force.
3. Finding the magnitude of the tangential acceleration of the car:
The tangential acceleration is the acceleration in the direction of the motion. In this case, since the car is moving along a circular path, the tangential acceleration is zero. This is because the velocity vector remains constant in magnitude but changes in direction.
4. Finding the total acceleration of the car:
To find the total acceleration, you need to combine the centripetal acceleration and the tangential acceleration. Since the tangential acceleration is zero in this case, the total acceleration will be equal to the centripetal acceleration.
I hope this explanation helps you solve the problem step by step. Remember to plug in the given values correctly and make sure your units are consistent throughout the calculations.