Question # 1 : What is the maximum acceleration a car can undergo if the coefficient of static friction between the tires and the ground is 0.82?
Question # 2 : A 1.5-kg block rests on top of a 7.5kg block. The cord and pulley have negligible mass, and there is no significant friction anywhere.
Part a) What force F must be applied to the bottom block so the top block accelerates to the right at 2.9 m/s^2 ?
Part b) What is the tension in the connecting cord?
Question #1: To determine the maximum acceleration a car can undergo, we need to use the coefficient of static friction between the car's tires and the ground. The maximum acceleration is limited by the maximum force of static friction between the tires and the ground.
The equation for maximum force of static friction can be written as:
F_friction = μ_static * N,
where F_friction is the maximum force of static friction, μ_static is the coefficient of static friction, and N is the normal force.
To find the maximum acceleration, we need to determine the maximum force of static friction. However, we don't have the value for the normal force. In most cases, the normal force is equal to the weight of the object on a flat surface. Assuming the car is on a flat surface, we can estimate the normal force using the weight of the car.
Therefore, to find the maximum acceleration (a_max), we'll use the equation:
a_max = F_friction / m,
where m is the mass of the car.
To solve this equation, follow these steps:
1. Determine the weight of the car. Multiply the mass of the car (in kg) by the acceleration due to gravity (approximately 9.8 m/s^2) to get the weight in Newtons.
2. Multiply the weight of the car by the coefficient of static friction (0.82) to find the maximum force of static friction (F_friction).
3. Divide the maximum force of static friction by the mass of the car to obtain the maximum acceleration (a_max).
Question #2 - Part a: To find the force F required to make the top block accelerate to the right at 2.9 m/s^2, we can use Newton's second law of motion.
The equation for Newton's second law is:
ΣF = ma,
where ΣF is the net force acting on an object, m is the mass of the object, and a is the acceleration of the object.
In this case, we have two blocks connected by a cord. Since there is no significant friction, the force applied to the bottom block will cause both blocks to move with the same acceleration.
To solve this equation, follow these steps:
1. Calculate the net force acting on the system of blocks by multiplying the total mass (mass of the top block + mass of the bottom block) by the desired acceleration (2.9 m/s^2).
2. The force applied to the bottom block (F) is the same as the net force acting on the system. Therefore, F = ΣF.
Question #2 - Part b: To find the tension in the connecting cord, we can analyze the forces acting on the top block in the vertical direction.
Since there is no significant friction and the pulley and cord have negligible mass, the tension in the cord will be the same throughout.
To find the tension, follow these steps:
1. Draw a free body diagram for the top block and identify all the forces acting on it. These forces include the weight of the top block (mg) and the tension in the cord (T).
2. Set up an equation based on Newton's second law of motion in the vertical direction:
ΣF_vert = ma_vert.
The sum of the forces in the vertical direction is the tension in the cord minus the weight of the top block (T - mg).
3. Solve the equation for the tension (T). Rearrange the equation to isolate T, and substitute the given values for the mass of the top block (1.5 kg) and the acceleration (you mentioned 2.9 m/s^2 in Part a).
By following these steps, you can determine the force required and the tension in the connecting cord.