1. A 1200 kg car has a maximum power output of 180 hp. How steep a hill can it climb at a constant speed of 106 km/h if the frictional forces add up to 600 N?
2. A 2.9 kg block slides with a speed of 1.1 m/s on a frictionless, horizontal surface until it encounters a spring.
(a) If the block compresses the spring 4.5 cm before coming to rest, what is the force constant of the spring?
N/m
(b) What initial speed should the block have if it is to compress the spring by 1.2 cm?
m/s
To answer the first question, we need to determine the maximum force that the car can produce at its maximum power output and then calculate the incline the car can climb.
1. First, convert the power from horsepower (hp) to watts (W). Since 1 horsepower is equal to 745.7 watts, the car's maximum power output is:
180 hp * 745.7 W/hp = 134,226 W
2. We can use the equation for power to find the maximum force:
Power = Force * Speed
Rearranging the equation, we get:
Force = Power / Speed
Substituting in the known values:
Force = 134,226 W / (106 km/h * 1000 m/km * 1/3600 h/s)
Force = 3,114 N
3. Now, subtract the frictional forces (600 N) to find the net force available for climbing the hill:
Net Force = Force - Frictional Forces
Net Force = 3,114 N - 600 N
Net Force = 2,514 N
4. The force required to climb an incline can be calculated using the equation:
Force = Mass * Gravity * sin(theta)
Rearranging the equation to solve for the angle (theta):
theta = arcsin(Force / (Mass * Gravity))
Substituting the known values:
theta = arcsin(2,514 N / (1,200 kg * 9.8 m/s^2))
theta ≈ 29.8 degrees
Therefore, the car can climb a hill with a maximum angle of approximately 29.8 degrees at a constant speed of 106 km/h, considering the given parameters.
Moving on to the second question:
(a) To find the force constant of the spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. The equation is given as:
Force = -k * x
Where:
Force is the force exerted by the spring
k is the force constant of the spring
x is the displacement of the block from the equilibrium position (compressed length)
Rearranging the equation to solve for the force constant (k):
k = -Force / x
Substituting the known values:
k = -m * acceleration / x
Since the block comes to rest and has zero acceleration, we can simplify the equation:
k = 0 / x
k = 0
Therefore, the force constant of the spring is 0 N/m.
(b) To find the initial speed required to compress the spring by 1.2 cm, we can use the conservation of mechanical energy.
The potential energy stored in a compressed spring is given by:
Potential Energy = (1/2) * k * x^2
Setting the potential energy equal to the initial kinetic energy of the block:
(1/2) * m * v^2 = (1/2) * k * x^2
Rearranging the equation to solve for the initial speed (v):
v^2 = (k/m) * x^2
v = sqrt((k/m) * x^2)
Substituting the known values:
v = sqrt((0 N/m) / (2.9 kg) * (0.012 m)^2)
v = 0 m/s
Therefore, the initial speed required to compress the spring by 1.2 cm is 0 m/s.