2.A ball rolls off a shelf with a horizontal velocity of v. At what horizontal distance from the shelf does the ball land if the height of the shelf is h above the floor?
3. A car travels around a circle with a diameter of 500 m at a constant speed of 25 m/s. The static friction coefficient is 0.3 and the kinetic friction coefficient is 0.2. Will the car skid?
To find the horizontal distance from the shelf where the ball lands, we can use the equation of motion in the horizontal direction. Since the ball rolls off the shelf with a horizontal velocity v and there are no horizontal forces acting on the ball (ignoring air resistance), the horizontal motion is uniform. Therefore, the horizontal distance (d) is given by the equation:
d = v * t
where t is the time it takes for the ball to reach the ground. To find t, we need to consider the vertical motion of the ball.
Since the ball is released from rest at a height h, its initial vertical velocity (u) is 0 m/s. Using the equation of motion in the vertical direction, we can find the time it takes for the ball to hit the ground:
h = u * t + (1/2) * g * t^2
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Rearranging the equation, we get:
(1/2) * g * t^2 = h
Simplifying further:
t^2 = (2h) / g
t = sqrt((2h) / g)
Now that we have the time, we can substitute it back into the equation for the horizontal distance:
d = v * t
Substituting the expression for t:
d = v * sqrt((2h) / g)
Simplifying the equation will give us the horizontal distance from the shelf where the ball lands.
For the second question, to determine if the car will skid while traveling around the circle, we need to compare the maximum static friction force with the force required to keep the car moving in a circle.
The maximum static friction force (Fs) can be calculated using the formula:
Fs = μs * N
where μs is the static friction coefficient and N is the normal force. In this case, the normal force is equal to the weight of the car, since the car is on a flat surface. The weight can be calculated using the formula:
Weight = mass * gravitational acceleration
Once we have the maximum static friction force, we can compare it to the centripetal force (Fc) required to keep the car moving in a circle. The centripetal force is given by:
Fc = (mass * velocity^2) / radius
where the radius is half the diameter of the circle.
If the maximum static friction force (Fs) is greater than or equal to the centripetal force (Fc), then the car will not skid and will be able to travel around the circle without slipping. However, if Fs is less than Fc, then the car will skid.