At what speed will a car round a 52-m-radius curve, banked at a 45 degree angle, if no friction is required between the road and tires to prevent the car from slipping? (g=9.8m/s^2)
To calculate the speed at which a car will round the curve, we can use the concept of the centripetal force required for circular motion. In this scenario, the force keeping the car in a curved path is the horizontal component of the car's weight.
Here's how we can calculate it step by step:
Step 1: Identify the forces acting on the car:
In this case, the only significant force acting on the car is the weight of the car (mg), where m is the mass of the car and g is the acceleration due to gravity.
Step 2: Determine the horizontal component of the weight:
Since the curve is banked at a 45-degree angle, the vertical force of gravity (mg) will have a component that is perpendicular to the curve's surface (mg * cos(45°)). However, this vertical component does not contribute to the car's curved motion. The horizontal component of the weight (mg * sin(45°)) will provide the centripetal force required to keep the car in a curved path.
Step 3: Equate the horizontal component of the weight to the centripetal force:
The centripetal force required for circular motion is given by the equation F = (mv^2) / r, where F is the centripetal force, m is the mass of the car, v is its velocity, and r is the radius of the curve. Since the horizontal component of the weight acts as the centripetal force, we can set them equal to each other:
mg * sin(45°) = (mv^2) / r
Step 4: Solve for the velocity (v):
Rearranging the equation to solve for v, we have:
v^2 = (g * r * sin(45°)) / 1
v^2 = (9.8 m/s^2 * 52 m * sin(45°)) / 1
v^2 = 363.04
Taking the square root of both sides, we get:
v = sqrt(363.04)
v ≈ 19.04 m/s
Therefore, the car will round the 52-meter-radius curve with a speed of approximately 19.04 m/s.
To find the speed at which the car will round the curve without slipping, we can use the concept of centripetal force.
Step 1: Determine the gravitational force acting on the car.
The gravitational force acting on the car is given by the formula: F_gravity = mass * gravitational acceleration (g),
where mass is the mass of the car and g is the acceleration due to gravity (9.8 m/s^2).
Step 2: Calculate the normal force.
The normal force, denoted by N, is the force exerted by the track on the car perpendicular to the surface. It can be found using trigonometry.
N = F_gravity * cos(angle),
where angle is the angle of banking (45 degrees).
Step 3: Determine the centripetal force required to keep the car on the curve.
The centripetal force, denoted by F_c, is given by the formula: F_c = (mass * velocity^2) / radius of the curve,
where mass is the mass of the car, velocity is the speed of the car, and radius of the curve is given as 52 m.
Step 4: Equate the centripetal force to the gravitational force.
F_c = F_gravity = N, since there is no friction.
Step 5: Solve for the velocity.
Substitute the values we have into the equation from step 3 and solve for velocity.
(mass * velocity^2) / radius of the curve = N
(mass * velocity^2) / 52 = N
(mass * velocity^2) / 52 = F_gravity * cos(angle)
(mass * velocity^2) / 52 = (mass * g) * cos(angle)
velocity^2 = 52 * g * cos(angle)
velocity = sqrt(52 * g * cos(angle))
Now, substitute the values of g (9.8 m/s^2) and angle (45 degrees) into the equation:
velocity = sqrt(52 * 9.8 * cos(45))
velocity ≈ 15.59 m/s
Therefore, the speed at which the car will round the 52-m-radius curve, banked at a 45 degree angle, without slipping is approximately 15.59 m/s.