For parts A-C:
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top view rear view
A car of mass, 2m, is going around a turn at a constant radius, R, with velocity, v0. The turn is not banked – it is a flat surface. Assume the car and the road are on Earth’s surface.
a) Is the friction that holds the car on the road around the turn static or kinetic? Explain your answer.
b) Find the minimum coefficient of friction needed for the car to make the turn in terms of v0, R, g, and any other variables or constants that are necessary for the expression.
c) Find the kinetic energy of the car in terms of m, v0, and any constants.
For parts D-F:
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d) The car hits an icy patch and careens off the turn in a straight line. It maintains its velocity, v0, until it hits a block of mass, m, attached to a massless spring with spring constant, k. The car has a perfectly inelastic collision with the block attached to the spring. Find the velocity, v2, of the combined car and block immediately after the collision takes place in terms of m, v0, and any needed constants.
e) Determine the distance that the spring compresses after the collision in terms of m, k, v0, and any necessary constants.
f) Find the time for the spring to return to its original position the first time after it compresses and starts to stretch in terms of m, k, and any other necessary variables or constants.
What happens to your answer in part f if the car has a mass of 11m instead?
a) The friction that holds the car on the road around the turn is static friction. This is because the car is not sliding or skidding, but rather rolling without slipping. Static friction occurs when two surfaces are in contact and not sliding relative to each other. In this case, the static friction between the tires of the car and the road provides the necessary centripetal force to keep the car moving in a curved path.
b) To find the minimum coefficient of friction needed for the car to make the turn, we can use the concept of centripetal force. The centripetal force required to keep the car moving in a circular path of radius R is given by the equation:
F_centripetal = m * v0^2 / R
The frictional force between the tires and the road provides the centripetal force. The maximum static frictional force is given by:
F_friction_max = μ_static * m * g
where μ_static is the coefficient of static friction and g is the acceleration due to gravity. For the car to make the turn, the maximum static frictional force should be equal to the centripetal force required:
μ_static * m * g = m * v0^2 / R
Simplifying the equation, we can find the minimum coefficient of friction:
μ_static = v0^2 / (g * R)
c) The kinetic energy of the car can be calculated using the formula:
KE = (1/2) * m * v0^2
where m is the mass of the car and v0 is the velocity of the car.
d) After the collision, the car and the block attached to the spring become a combined system. The velocity of the combined system, v2, can be found using the principle of conservation of momentum. Before the collision, the momentum of the car is given by:
p_car = m * v0
The momentum of the block attached to the spring is initially zero since it is at rest. After the collision, the combined system has a total momentum given by:
p_combined = (m + m) * v2 = 2m * v2
Applying the conservation of momentum, we have:
p_car = p_combined
m * v0 = 2m * v2
Simplifying the equation, we get the velocity of the combined system after the collision:
v2 = v0 / 2
e) The distance that the spring compresses after the collision can be found using the conservation of mechanical energy. Since the collision is perfectly inelastic, the kinetic energy before the collision is transformed into potential energy stored in the compressed spring. The initial kinetic energy of the car before the collision is given by:
KE_initial = (1/2) * m * v0^2
The potential energy stored in the compressed spring is given by:
PE_spring = (1/2) * k * x^2
where k is the spring constant and x is the distance the spring compresses.
Applying the conservation of mechanical energy, we have:
KE_initial = PE_spring
(1/2) * m * v0^2 = (1/2) * k * x^2
Solving for x, the distance the spring compresses, we get:
x = √((m * v0^2) / k)
f) The time for the spring to return to its original position the first time after it compresses and starts to stretch can be found using the principles of simple harmonic motion. The period of oscillation, T, is given by:
T = 2π * √(m / k)
where m is the mass and k is the spring constant.
If the car has a mass of 11m instead, the equation for the period of oscillation in part f doesn't change as it only depends on the mass and spring constant, not the car mass. Therefore, the answer in part f will remain the same.