1. what happens to the force required for an object to maintain circular motion if the period is halved (goes twice as fast around the circle)?

a) force doubles
b) force quadruples
c) force is halved
d) force is quartered

2. In a pulley problem involving an inclined plane:

a) the gravity need to be resolved into components
b) the tension need to be resolved into components
c) the friction need to be resolved into components
d) the direction convention is the same for both blocks

3. Which of the following statements is true:

a) a acceleration of free fall objects on Earth is approximately 9.8 m/s^2 [down]
b) heavier objects in free fall accelerate faster than light objects
c) in projectile motion, the initial speed is always equal to the final speed
d) none of the statements are true

1. To understand what happens to the force required for an object to maintain circular motion if the period (time to complete one revolution) is halved, we can use the concept of centripetal force. The centripetal force is the force directed toward the center of the circle that keeps an object in circular motion.

The formula for centripetal force is F = (m * v^2) / r, where F is the centripetal force, m is the mass of the object, v is the velocity of the object, and r is the radius of the circle.

When the period is halved, it means the object completes each revolution in half the time. This indicates that the object is moving faster around the circle.

To find the effect on the force required, we can analyze the formula. When the velocity of the object increases (as it moves faster), the square of the velocity (v^2) increases at a faster rate. So, if the velocity doubles, the v^2 term will be four times larger.

In this case, since the object is moving twice as fast around the circle, the velocity (v) becomes 2v. Substituting this into the formula, we get F = (m * (2v)^2) / r = 4 * (m * v^2) / r.

Therefore, we can conclude that when the period is halved and the object is moving twice as fast, the force required to maintain circular motion quadruples. Hence, the correct answer is b) force quadruples.

2. In a pulley problem involving an inclined plane, there are several components that need to be considered.

First, we need to resolve the force of gravity acting on the objects into its components. The force of gravity usually has two components: one parallel to the inclined plane and one perpendicular to it.

Next, we need to resolve the tension in the rope (or cable) into components. The tension is the force transmitted along the rope and will depend on the situation and the angles involved. It can be resolved into components parallel and perpendicular to the inclined plane.

Third, we need to consider the friction force, which also acts on the inclined plane. Depending on the direction of motion and the coefficient of friction, the friction force may need to be resolved into components.

Lastly, it is important to define a consistent direction convention for both blocks and any forces involved, so that calculations and analysis can be done accurately.

Therefore, the correct answer is a) the gravity needs to be resolved into components.

3. Among the given statements, "a acceleration of free fall objects on Earth is approximately 9.8 m/s^2 [down]" is true. When an object falls freely under the influence of gravity on Earth, it experiences an acceleration of approximately 9.8 m/s^2 directed downward. This is often denoted as g, which represents the acceleration due to gravity near the Earth's surface.

The other statements are false:
- Heavier objects and lighter objects accelerate at the same rate in free fall, which means that they accelerate at approximately 9.8 m/s^2.
- In projectile motion, the initial speed and the final speed are not always equal. The final speed will depend on factors such as the launch angle, air resistance, and height of the projectile.
- Therefore, the correct answer is d) none of the statements are true.