Problem 46 - College Physics

Question

You are driving with a friend who is sitting to your right on the passenger side of the front seat of your car. You would like to be closer to your friend, so you decide to use physics to achieve your romantic goal by making a quick turn. (a) Which way (to the left or the right) should you turn the car to get your friend to slide toward you? (b) If the coefficient of static friction between your friend and the car seat is 0.55 and you keep driving at a constant speed of \(15 \mathrm{m} / \mathrm{s},\) what is the maximum radius you could make your turn and still have your friend slide your way?

Step-by-Step Solution

Answer

(a) Turn the car to the left, (b) Maximum radius is approximately 41.68 meters.

Step 1: Understand the Problem

To determine the direction in which to turn the car so that your friend slides towards you, we need to consider the physics of circular motion. The companion will slide due to the lack of sufficient centripetal friction if the static friction is overcome.

Step 2: Determine the Direction of Turn

To make your friend slide towards you, you need your friend to continue in a straight line due to inertia, while the car (and you) turn. Thus, if your friend is to your right, you should turn the car to the left. This way, your friend's inertia will cause them to slide towards you.

Step 3: Apply Centripetal Force Concept

Maximum static friction provides the necessary centripetal force to keep your friend from sliding. Static friction force is given by \(f_s = \mu_s \times m \times g\), and it must be greater than the centripetal force \(f_c = \frac{m v^2}{r}\) for your friend to not slide. Here, \(\mu_s\) is the static friction coefficient, \(m\) is the mass of your friend, \(g\) is acceleration due to gravity, \(v\) is the velocity, and \(r\) is the radius of the turn.

Step 4: Solve for Maximum Radius

Equalize the maximum static friction force to the required centripetal force for sliding: \(\mu_s \times m \times g = \frac{m v^2}{r}\). Cancel out \(m\) (mass of friend) from both sides and solve for \(r\): \[\mu_s \times g = \frac{v^2}{r}\].Rearrange to find \(r\):\[r = \frac{v^2}{\mu_s \times g}\].Substitute the known values: \(v = 15 \text{ m/s}\), \(\mu_s = 0.55\), \(g = 9.81 \text{ m/s}^2\).\(r = \frac{(15)^2}{0.55 \times 9.81}\).

Step 5: Calculate the Maximum Radius

Compute the values:\(\mu_s \times g = 0.55 \times 9.81 = 5.3955 \text{ m/s}^2\).Substitute into the equation for \(r\):\[r = \frac{225}{5.3955} \approx 41.68 \text{ meters}\].

Key Concepts

Static Friction

In the context of our problem, static friction is the force that keeps your friend stationary relative to the car seat while you are driving. It's a type of friction that acts between surfaces that are not moving relative to each other. Think of it as a "holding" force. As you drive, static friction helps keep your friend from sliding across the seat during normal conditions.

How Static Friction Works

Its magnitude depends on two main factors:

• the coefficient of static friction (\(\mu_s\)) which is a dimensionless constant representing the sticky nature between the two surfaces (i.e., your friend and the car seat)
• the normal force, which in this case, is primarily your friend's weight pushing down on the seat

The force of static friction can be calculated using the formula:\[f_s = \mu_s \times m \times g\]where:

• \(m\) is the mass of your friend
• \(g\) is the acceleration due to gravity
• \(f_s\) is the static friction force

When you decide to make a turn, if the force trying to slide your friend across is greater than this static friction force, they will start to move on the seat.

Centripetal Force

Centripetal force is crucial in circular motion scenarios like turning a car. It acts towards the center of a curve or circular path, keeping objects moving along their circular route. In our scenario, as you turn the car, this force needs to be strong enough to counteract inertia, which prefers your friend to go straight.

The Role of Centripetal Force in Turning

The important thing to note is that in a turn, this force can come from different sources:

• The road's reaction to the wheels' force
• Static friction between your friend and the car seat as they want to resist sliding

The actual formula used to calculate the necessary centripetal force (\(f_c\)) is:\[f_c = \frac{m v^2}{r}\]where:

• \(m\) is the mass of your friend
• \(v\) is the velocity of the car
• \(r\) is the radius of the path

In practical terms, if the required centripetal force exceeds the static friction force, sliding will occur. Thus, maximizing the radius of curvature will help maintain a balance and prevent sliding.

Inertia

Inertia can be thought of as the natural tendency of an object to resist changes in its state of motion. This concept is one of Newton's foundational principles, indicating that anything in motion stays in motion unless acted upon by a force.

Understanding Inertia's Impact

When your car makes a turn to the left, your friend naturally wants to continue in the direction they were traveling—as per their inertia. That's why it's said they slide to the left—or towards you—if the turn is sharp enough.

Implications in Circular Motion Scenarios

In our driving example:

• This inertia leads to your friend resisting the car's change in direction. It's like their body wishes to proceed in a straight line.
• The turn adds a centripetal force to alter this motion, which needs to be strong enough to counter inertia but not exceed static friction to prevent sliding.

To wrap it up, understanding inertia helps make sense of why objects resist motion changes and is key to controlling slide or turns in your car effectively.