If the coefficient of static friction between a car’s tires and the pavement is 0.65, calculate the minimum torque that must be applied to the 66-cm-diameter tire of a 1250-kg automobile in order to “lay rubber” (make the wheels spin, slipping as the car accelerates). Assume each wheel supports an equal share of the weight.

Verified step by step guidance

1. Calculate the normal force acting on each tire. Since the weight of the car is evenly distributed among all four tires, divide the total weight of the car by four. Use the formula for weight, $W=mg$, where $m$ is the mass of the car and $g$ is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Determine the maximum static friction force that can act on each tire using the formula $f_{s,max}={\mu }_{s}N$, where ${\mu }_s$ is the coefficient of static friction and $N$ is the normal force calculated in the previous step.
3. Calculate the radius of the tire from its diameter. Since the diameter is given as 66 cm, convert this measurement to meters and then divide by 2 to find the radius.
4. Use the relationship between torque ($\tau$), radius ($r$), and frictional force ($f_{s,max}$) to find the minimum torque needed to overcome the static friction. The formula is $\tau =f_{s,max}\times r$.
5. Apply the calculated torque to each tire to determine the total torque required to make all wheels spin, considering that each wheel needs the same amount of torque to start slipping.

Key Concepts

Static Friction

Static friction is the force that resists the initiation of sliding motion between two surfaces in contact. It is characterized by a coefficient, which in this case is 0.65, indicating the ratio of the maximum static friction force to the normal force. This force is crucial for determining how much torque is needed to overcome the friction and initiate wheel spin.

Torque

Torque is a measure of the rotational force applied to an object, calculated as the product of the force and the distance from the pivot point (radius). In this scenario, the torque applied to the tire must be sufficient to overcome the static friction force to cause the wheels to spin. The formula for torque is τ = r × F, where τ is torque, r is the radius of the tire, and F is the force applied.

Weight Distribution

Weight distribution refers to how the weight of the automobile is shared among its wheels. In this case, the total weight of the car (1250 kg) is evenly distributed across the four tires. This distribution affects the normal force on each tire, which in turn influences the maximum static friction force that can be generated, impacting the torque calculation needed to make the wheels slip.