Problem 73 - College Physics

Question

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 \(750 \mathrm{~kg}\), the radius of that section of the track is \(55.0 \mathrm{~m},\) and the car entered the bottom after descending vertically (from rest) \(25.0 \mathrm{~m}\) on a frictionless straight incline. If the braking force is a steady \(1700 \mathrm{~N}\), determine (a) the car's centripetal acceleration (including direction), \((\mathrm{b})\) the normal force of the track on the car, (c) the tangential acceleration of the car (including direction), and (d) the total acceleration of the car.

Step-by-Step Solution

Answer

(a) 8.86 m/s² (towards center), (b) 13995 N, (c) 2.27 m/s² (upwards), (d) 9.14 m/s².

Step 1: Calculate the car's speed at the bottom of the incline

First, find the speed of the car at the very bottom using conservation of energy. The potential energy at the top transforms into kinetic energy at the bottom. \[ mgh = \frac{1}{2} mv^2 \]Given: mass \( m = 750\, \mathrm{kg} \), height \( h = 25.0\, \mathrm{m} \), \( g = 9.8\, \mathrm{m/s^2} \). Solve for \( v \) : \[ v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 25.0} \approx 22.1\, \mathrm{m/s} \].

Step 2: Calculate the centripetal acceleration

The centripetal acceleration formula is \[ a_c = \frac{v^2}{r} \]Substitute \( v = 22.1\, \mathrm{m/s} \) and radius \( r = 55.0\, \mathrm{m} \) :\[ a_c = \frac{(22.1)^2}{55} \approx 8.86\, \mathrm{m/s^2} \]\The direction of the centripetal acceleration is towards the center of the semicircular path.

Step 3: Determine the normal force exerted by the track

At the bottom of the loop, the forces include gravity and normal force. Use Newton's second law in the radial direction:\[ N - mg = \,m a_c \]\[ N = m(g + a_c) \]Substitute \( m = 750\, \mathrm{kg} \), \( g = 9.8\, \mathrm{m/s^2} \), \( a_c = 8.86\, \mathrm{m/s^2} \): \[ N = 750(9.8 + 8.86) = 750 \times 18.66 \approx 13995\, \mathrm{N} \].

Step 4: Calculate the tangential acceleration

The braking force is acting against the motion, providing tangential deceleration. Use:\[ F = ma_t \Rightarrow a_t = \frac{F}{m} \]\[ a_t = \frac{1700}{750} \approx 2.27\, \mathrm{m/s^2} \]The direction is opposite to the velocity, i.e., "up the incline".

Step 5: Calculate the total acceleration

Using vector addition for the centripetal (radial) and tangential accelerations:\[ a_{\text{total}} = \sqrt{a_c^2 + a_t^2} \]\[ a_{\text{total}} = \sqrt{(8.86)^2 + (2.27)^2} \approx 9.14\, \mathrm{m/s^2} \]Direction is a combination "towards the center of the curve and upwards against motion".

Key Concepts

Conservation of Energy

In the realm of physics, the conservation of energy principle is a fundamental concept that ensures energy within an isolated system remains constant over time. When dealing with roller coasters or similar systems, this principle simplifies how we analyze the motion of objects. Here, as the roller coaster descends from a height, its potential energy is converted to kinetic energy. The transformation adheres to the equation:\[ mgh = \frac{1}{2} mv^2\]where:

• \(m\) is the mass of the roller coaster (750 kg in this case),
• \(g\) is the acceleration due to gravity (9.8 m/s²),
• \(h\) is the height (25 m),
• \(v\) is the velocity at the base of the slope.

By solving for \(v\), the velocity at the lowest point of the incline, we find it to be approximately 22.1 m/s. This equation illustrates how potential energy (due to height) is completely transformed into kinetic energy (associated with motion) when other forces such as friction are negligible.

Newton's Second Law

Newton's Second Law is crucial in understanding the forces involved in motion, stating that the acceleration of an object is directly proportional to the net force acting upon it, and inversely proportional to its mass. It can be expressed as:\[ F = ma \]where:

• \(F\) is the net force applied,
• \(m\) is the mass,
• \(a\) is the acceleration.

In the context of our roller coaster scenario at the bottom of the descent, the normal force exerted by the track needs to counterbalance the gravitational force while also providing the centripetal force necessary due to the circular motion:\[ N - mg = ma_c \]Solving for the normal force \(N\) includes both gravitational and centripetal components. Newton's Second Law here helps determine that the normal force provided by the track is nearly 13,995 N, a force reflecting both the support against gravity and the need to maintain circular motion.

Braking Force

The braking force applies a crucial role in the modification of a roller coaster's motion. As soon as the emergency brake is engaged, a steady force of 1700 N opposes the direction of motion. This force results in what we call tangential deceleration:\[ F = ma_t \]resulting in:\[ a_t = \frac{F}{m} = \frac{1700}{750} \approx 2.27 \, \text{m/s}^2 \]The direction of this force is opposite to the motion, effectively slowing down the roller coaster. It acts along the path of travel, which in this scenario, is "up the incline." Understanding braking force is essential for ensuring the safety and proper functioning of rides, as it directly affects the kinetic state and, consequently, the experience and safety of passengers.

Tangential Acceleration

Tangential acceleration refers to the change in speed of an object moving along a curved path. It acts along the direction of the velocity, influencing the speed, not the direction of travel. In the case of our roller coaster, tangential acceleration is particularly significant because it's affected by the braking force.The calculation of tangential acceleration here is obtained by:\[ a_t = \frac{F}{m} \]with \(F\) as the braking force and \(m\) as the mass of the roller coaster. This equation gives the rate at which the roller coaster slows along the track due to the braking system. Since the braking force is against the motion, the tangential acceleration is negative relative to the original direction of motion, which implies deceleration. By understanding and calculating tangential acceleration, engineers and physicists can predict and manage how quickly rides can safely slow down, while ensuring the thrill isn't negatively impacted.